# 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: