Question 7.6.1: Temperature Dynamics of a Mixing Process Liquid at a tempera...
Temperature Dynamics of a Mixing Process
Liquid at a temperature T_{i} is pumped into a mixing tank at a constant volume flow rate q_{v} (Figure 7.6.1). The container walls are perfectly insulated so that no heat escapes through them. The container volume is V, and the liquid within is well mixed so that its temperature throughout is T . The liquid’s specific heat and mass density are c_{p} and ρ. Develop a model for the temperature T as a function of time, with T_{i} as the input.
The amount of heat energy in the tank liquid is ρ c_{p} V(T − T_{r}), where T_{r} is an arbitrarily selected reference temperature. From conservation of energy,
\frac{d [ρ c_{p} V(T − T_{r}) ]}{d t} = heat rate in − heat rate out (1)
Liquid mass is flowing into the tank at the rate \dot{m} = ρ q_{v}. Thus heat energy is flowing into the tank at the rate
heat rate in = \dot{m} c_{p}(T_{i} − T_{r}) = ρ q_{v} c_{p}(T_{i} − T_{r})
Similarly,
heat rate out = ρ q_{v} c_{p}(T − T_{r})
Therefore, from equation (1), since ρ, c_{p}, V, and T_{r} are constants,
ρ c_{p} V \frac{d T}{d t} = ρ q_{v} c_{p}(T_{i} − T_{r}) − ρ q_{v} c_{p} (T − T_{r}) = ρ q_{v} c_{p}(T_{i} − T )
Cancel ρ c_{p} and rearrange to obtain
\frac{V}{q_{v}} \frac{d T}{d t} + T = T_{i}
Note that T_{r}, ρ, and c_{p} do not appear in the final model form, so their specific numerical values are irrelevant to the problem. The system’s time constant is V/q_{v}, and thus the liquid temperature T changes slowly if the tank volume V is large or if the inflow rate q_{v} is small, which agrees with our intuition.