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.

 

[Nickie]

Hi there,
Your attempt at a solution is a good start, but there are a few things to consider. Firstly, your equations for mass and heat transfer are correct, but your equation for the change in temperature with time is not quite right. It should be dT/dt = (Tin - T(t))/τ0, as the temperature in the container is changing towards the temperature of the incoming fluid, not towards the initial temperature T0. Also, your equation for T(t) should be T(t) = Tin + (T0 - Tin)e^-t/τ0, as the temperature in the container is increasing towards the incoming temperature Tin, not decreasing. Finally, your equation for the time at which the temperature will be T should be t = τ0ln((T - Tin)/(T0 - Tin)), as the temperature in the container is increasing towards Tin, not towards T0. Overall, your approach is correct, but there are some minor errors that need to be corrected to get the correct solution. Keep up the good work!