I am given an equation of the form : dT/dt* = -lambda*(T-T_inf) + K.d^2T/dx*^2 over 0<x*<inf , t*>0
With initial condition: T(x*,0) = T_inf + T0[1-(x*/L)^2] , 0<= x* <= L
and T(x*,0) = T_inf , x*>L
I need to nondimensionalise the equation using the scaled variables:
t = Kt*/L^2, x = x*/L, theta = (T-T_inf)/T0
My attempt at nondimensionalising is as follows:
d^2/dx*^2 = 1/L^2 * d^2/dx^2
d/dt* = K/L^2 * d/dt
and
d^2T/dx*^2 = 1/L^2 * d^2T/dx^2 = 1/L^2 * T0 * dtheta/dx
dT/dt* = K/L^2 * dT/dt = K/L^2 * T0 * dtheta/dt
which I then substitute into the original equation to get:
K/L^2 * T0 * dtheta/dta = -lambda*(T-T_inf) + KT0/L^2 * d^2theta/dx^2
which leaves:
dtheta/dt = -lambda*L^2/K * theta + d^2theta/dx^2
and we are told to set y = lambda*L^2/K so
dtheta/dt = -y*theta + d^2theta/dx^2
Is this correct? I have had trouble in the past nondimensionalising so I am not sure. Additionally, I am unsure how to solve this equation. We were only taught how to solve the nonhomogeneous heat eqn with a source term in either x or t but not x,t which I am assuming this is (theta(x,t)) which makes me think I am wrong. I'd also appreciate help solving this regardless, source terms make it hard! Thanks!
edit: I'd like to note that I may have misinterpreted the question in terms of having to solve it. The subsequent question says that the internal heat of the system Q(t) = int[0,inf.] theta(x,t) dx and show that the heat is given by Q = Q0*e^(-yt)
With initial condition: T(x*,0) = T_inf + T0[1-(x*/L)^2] , 0<= x* <= L
and T(x*,0) = T_inf , x*>L
I need to nondimensionalise the equation using the scaled variables:
t = Kt*/L^2, x = x*/L, theta = (T-T_inf)/T0
My attempt at nondimensionalising is as follows:
d^2/dx*^2 = 1/L^2 * d^2/dx^2
d/dt* = K/L^2 * d/dt
and
d^2T/dx*^2 = 1/L^2 * d^2T/dx^2 = 1/L^2 * T0 * dtheta/dx
dT/dt* = K/L^2 * dT/dt = K/L^2 * T0 * dtheta/dt
which I then substitute into the original equation to get:
K/L^2 * T0 * dtheta/dta = -lambda*(T-T_inf) + KT0/L^2 * d^2theta/dx^2
which leaves:
dtheta/dt = -lambda*L^2/K * theta + d^2theta/dx^2
and we are told to set y = lambda*L^2/K so
dtheta/dt = -y*theta + d^2theta/dx^2
Is this correct? I have had trouble in the past nondimensionalising so I am not sure. Additionally, I am unsure how to solve this equation. We were only taught how to solve the nonhomogeneous heat eqn with a source term in either x or t but not x,t which I am assuming this is (theta(x,t)) which makes me think I am wrong. I'd also appreciate help solving this regardless, source terms make it hard! Thanks!
edit: I'd like to note that I may have misinterpreted the question in terms of having to solve it. The subsequent question says that the internal heat of the system Q(t) = int[0,inf.] theta(x,t) dx and show that the heat is given by Q = Q0*e^(-yt)
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