With β=        and γ =         . In order to estimate f (t) we take a look at the simulation result for this function which is a position derivative of T(x, t) at the end of radiator. It turns out we can approximate f (t) with an exponential function roughly as shown in Fig.5
We know the initial and final value of f (t). Also, the minimum of f (t) occurs at the transportation time of flow to the end of radiator i.e.         .  Therefore, we approximate f (t) as bellow:
 
With f0 = -   _ (Tin - Ta)             , f1 = -       (Tin - Ta)               and τ =              .
 
Fig.5. Simulation results for scaled Tout (t), its first position derivative and its approximation are shown. The first position derivative i.e. f (t) is approximated with an exponential function..
Substituting f (t) in (18), the return temperature is obtained as follows:
 
With c0 = Ta-       , c2 =                       and c1 = Tout, 0 - c0 - c2.
Back to (17), we substitute Tout (t) in the equation. Q (t) becomes:
 
The result is not a precise solution because we have made an approximation while deriving Tout (t). But it is still enough for us to extract useful information regarding the time constant and gain. The analytic solution and simulation for a specific flow rate is shown in Fig.6.
 
Fig.6.Simulation and analysis results for Q(t). The analytic solution gives us a good enough approximation of the transient and final behavior of the radiator output heat. We utilize this analytic solution to extract the parameters of a first order approximation of Q (t) step response.
The overshoot in the analytic solution compared to the simulation is due to neglecting an undershoot in Tout (t) calculations.
In the next section, we utilize the derived formula to extract the required gain and time constant for the control oriented LPV model.
D.    Radiator LPV Model
Parameters Krad and τrad of the radiator LPV model (7) are derived based on first order approximation of the radiator power step response (22). The steady state gain is:
 
With Tout, 1 corresponding to the flow rate q1. Using the tangent to Q (t) at t = 0 we can obtain the time constant. The slope of the tangent would be equivalent to the first derivative of Qfinal + (Q0 - Qfinale)                   at t = 0 which gives:
 
Therefore, at a specific operating point, the radiator gain and time constant can be obtained via (24) and (23). For a set of operating points these parameters are shown as two profiles of curves in Fig. 7.
 
Fig.7. Steady state gain and time constant variations for various values of the radiator flow and room temperature. The arrows show the direction of room temperature increase. Room temperature is changed between -10℃and 24℃ and flow is changed between the minimum and maximum flow
Fig. 7 shows that the radiator gain and time constant of the heat-flow transfer function significantly depend on the flow rate. The high gain and the long time constant in the low heat demand conditions mainly contribute to the oscillatory behavior. The control oriented model of room-radiator can be written as:
 
Room parameters, Ka and τa can be estimated easily by performing a simple step response experiment. We obtained these parameters based on [2] assuming specific materials for the components. 散热器恒温阀英文文献和中文翻译(4):http://www.youerw.com/fanyi/lunwen_13992.html