- #1
member 428835
Hi PF!
Given: ##u_t = u_{xx} +1## (heat equation) with the following B.C.: ##u_x(0,t)=1, u_x(L,t)= B, u(x,0)=f(x)##. My professor then continued by stating that in equilibrium, we have ##0 = u_{xx} +1 \implies u = -x^2/2 + C_1 x + C_2##. So far I'm on board, although by "equilibrium" does he mean steady state (after a long period of time)?
Next, we show that the equilibrium solution must satisfy ##-L + 1 = B## (via the B.C., after showing that ##C_1 = 1##).
He goes on to say that this condition is imposed from the source of heat and the two flux B.C. By source, does he mean the ##1## on the R.H.S of the initial equation?
Now here's where it get's tricky. He then says to perform an energy balance, thus:
$$\frac{d}{dt} \int_0^L c \rho u dx = -u_x(0) + u_x(L) + \int_0^L Q_0 dx = -1 +B +L=0$$
Now I understand the L.H.S completely, but where is he getting the heat source on the R.H.S, namely the ##Q_0## term? Also, shouldn't the thermal constant ##k## show up prefixing the flux terms (the ##u_x## terms) or is this only true if we applied the divergence theorem and fouriers law of heat? Also, why is there a minus sign in front of the ##u_x(0)## term and a positive sign in front of the ##u_x(L)##? Isn't it energy in minus energy out?
He then proceeds by stating that initial energy equals final energy in equilibrium, thus we must have $$\int_0^L f(x) dx = \int_0^L u(x) dx$$ where i know ##f(x)## is initial but is ##u(x)## final since it does not depend on ##t##?
thanks a ton!
Given: ##u_t = u_{xx} +1## (heat equation) with the following B.C.: ##u_x(0,t)=1, u_x(L,t)= B, u(x,0)=f(x)##. My professor then continued by stating that in equilibrium, we have ##0 = u_{xx} +1 \implies u = -x^2/2 + C_1 x + C_2##. So far I'm on board, although by "equilibrium" does he mean steady state (after a long period of time)?
Next, we show that the equilibrium solution must satisfy ##-L + 1 = B## (via the B.C., after showing that ##C_1 = 1##).
He goes on to say that this condition is imposed from the source of heat and the two flux B.C. By source, does he mean the ##1## on the R.H.S of the initial equation?
Now here's where it get's tricky. He then says to perform an energy balance, thus:
$$\frac{d}{dt} \int_0^L c \rho u dx = -u_x(0) + u_x(L) + \int_0^L Q_0 dx = -1 +B +L=0$$
Now I understand the L.H.S completely, but where is he getting the heat source on the R.H.S, namely the ##Q_0## term? Also, shouldn't the thermal constant ##k## show up prefixing the flux terms (the ##u_x## terms) or is this only true if we applied the divergence theorem and fouriers law of heat? Also, why is there a minus sign in front of the ##u_x(0)## term and a positive sign in front of the ##u_x(L)##? Isn't it energy in minus energy out?
He then proceeds by stating that initial energy equals final energy in equilibrium, thus we must have $$\int_0^L f(x) dx = \int_0^L u(x) dx$$ where i know ##f(x)## is initial but is ##u(x)## final since it does not depend on ##t##?
thanks a ton!