How to calculate the time to heat up a tank using recirculating liquid

Question

My question is about a tank filled with some liquid (properties of the liquid are known and can be considered constant. Specific heat equals 1.5 kJ/(kg.K)). The initial temperature of the liquid in the tank is 90 ºC. Now imagine that 2000 L/h of this liquid are recirculating, but before returning to the tank they are heated somehow (doesn't matter how) until they reach 240 ºC (assume it remains a liquid at this temperature).

How can I estimate how much time it would take before the tank temperature reaches 230 ºC. The liquid volume on the tank is constant and equal to 10000 L.

It's kind of an open problem (the values are only hypothetical so that the problem can be implement on Excel). How should I start analyzing it? What equations should I be looking for?

Answer 1

The mass balance gives mass flow in = mass flow out = $\dot{m}$. The energy balance on an insulated, constant volume tank gives

$$\dot{m}\tilde{C}_p(T - T_h) + m_T\tilde{C}_V\frac{dT}{dt} = 0 $$

where the variables are tank temperature $T$, hot temperature $T_h$, specific heat capacities $\tilde{C}_p$ and $\tilde{C}_V$, and tank fluid mass $m_T$. The ratio $m_T/\dot{m}$ is the residence time in the tank $t_R$. The residence time is also a ratio of tank volume to volumetric flow. Allow $\tau = t/t_R$ and $\theta = (T_h - T)/(T_h - T_o)$ where $T_o$ is the initial temperature in the tank. The differential equation becomes

$$\theta = \gamma \frac{d\theta}{d\tau} $$

with $\gamma = \tilde{C}_V / \tilde{C}_p$. When $\gamma = 1$, the answer follows with the boundary condition that $\theta = 1$ at $\tau = 0$.

$$\theta = \exp(-\tau) $$

An example plot is below.

Modifications for $\gamma \neq 1$ are left as an exercise.