Fluid's temperature change in time.

1. Dec 23, 2012

peripatein

Hello,
1. The problem statement, all variables and given/known data
The volume of the container in the attachment is given as V. The container is filled with a fluid whose heat capacity is C and whose viscosity decreases linearly with the temperature.
The fluid is initially at temperature T0, and a pipe carries fluid at temperature Tin into it and at a rate which is equal to the rate of the volume lost from the container (i.e. its dV/dt). I am asked to find the time at which the temperature in the container will be T.

2. Relevant equations

3. The attempt at a solution
M/V = ρ
m(t) = ρVt/τ0, where m(t) denotes the mass of water which passed through the pipe after time t.
dQ/dt = [ρV/τ0]*Cw(T(t) - Tc) = ρVCwdT/dt
Hence, dT/dt = (Tc - T(t))/T0
Hence, T(t) = Tc + (TH - Tc)e-t/τ0 = Tin + (T0 - Tin)e-t/τ0

t = -τ0ln((T - Tin)/(T0 - Tin))

Is that correct?

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2. Dec 23, 2012

Staff: Mentor

Well, the final answer looks right, but, astonishingly, some of the equations leading up to the final answer don't look right. If τ0 is the mean residence time in the tank, then the equation for m(t) should not contain a t, and m(t) itself should not be a function of t. Your equations contain a Tc, but nowhere is this parameter defined. It would appear that Tc is the same as Tin, but I can't understand why you deemed it necessary to introduce another parameter name. Otherwise, OK.

3. Dec 24, 2012

peripatein

Supposing I wished to express that answer without tau_0, how could I do so granted that the volume of the tank is V?

4. Dec 24, 2012

Staff: Mentor

τ0 = ρV/m, so just substitute that into your solution.