How to find the 'reaction speed' required in a valve for a certain application

I know this is answerable but I’m not sure I know where to start:

You have a pump (constant volume flow) supplying the primary side of a heat exchanger with hot water (flow). The temperature of the flow is adjusted by mixing in cold return with a three way mixer (valve). The temperature of the medium on the secondary side out of the HX $t_{so}$ is measured, as is the flow temperature (primary side in) $t_{pi}$. The goal is a specific value $T_{so}$ for $t_{so}$ (lowercase letters: actual measurements, uppercase letters: setpoints).

I see two principal control strategies (and within each the question if one uses PID controls or simpler schemes):

• control valve directly for $t_{s}$
• control valve for a given $T_{f}$, if $t_{s} > T_{s}$ for a given length of time, adjust $T_{f}$ downwards and vice versa

My question is: If there’s a sudden jumps in temperature or flow on the inlet of the secondary side – say 10K in 5 seconds, or flow drops to 75% within 2 s, and I have a certain parameters – say $t_s$ must never be higher then $T_s + 5K$ or $t_s = T_s +_- 0.5 K$ 60 s after the disturbance is registered, how do I know how fast acting the three way mixer must be (from 0-100% in 120s?)? The numbers are of course just indicative.

Analytically, the required $t_{pi}$, can be calculated iterativly using the following ($dot m$ mass flows on primary and secondary sides, $c$ respective heat capacities):

$$ t_{pi}=t_{po} – frac{dot m_s c_s}{dot m_p c_p} * (t_{so} – t_{si})$$

NMech provided a closed form:

$$ T_{p,i} = T_{s,o} – frac{
Q }{A k} ProductLog[-frac{A k(T_{p,o}-T_{s,i}) e^{-frac{A k }{Q}(T_{p,o}-T_{s,i}) } }{Q}]$$

Where:

• $ProductLog[x]$ is the Lambert W function.