Fluid's temperature change in time.

[peripatein]

Hello,

Homework Statement

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 T_in 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.

Homework Equations

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]*C_w(T(t) - T_c) = ρVC_wdT/dt
Hence, dT/dt = (T_c - T(t))/T₀
Hence, T(t) = T_c + (T_H - T_c)e^-t/τ₀ = T_in + (T₀ - T_in)e^-t/τ₀

t = -τ₀ln((T - T_in)/(T₀ - T_in))

Is that correct?

 

[Chestermiller]

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 T_c, but nowhere is this parameter defined. It would appear that T_c is the same as T_in, but I can't understand why you deemed it necessary to introduce another parameter name. Otherwise, OK.

 

[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?

 

[Chestermiller]

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