A process temperature sensor/transmitter in a fermentation

Step 1 of 4

Consider the two time constants of a second-order dynamic equation of a temperature sensor/transmitter which are \(\tau_{1}=1 \mathrm{\ s}\) and \(\tau_{2}=0.1 \mathrm{\ s}\).

Form a second-order dynamic equation using the time constants by considering the gain of the transfer function is one.

\(\frac{T_m^{\prime}(s)}{T^{\prime}(s)}=\frac{1}{\left(\tau_1s+1\right)\left(\tau_2s+1\right)}\ldots\ldots\left(1\right)\)

Here,

\(T_{m}^{\prime}(s)\) is the measured value of temperature.

\(T^{\prime}(s)\) is the temperature being measured.

Consider the following ramp temperature change:

\(T^{\prime}(t)=0.1 t\left({ }^{\circ} \mathrm{C} / \mathrm{s}\right) \ldots . .(2)\)

Take Laplace transform for equation (2).

\(\begin{array}{l} L\left[T^{\prime}(t)\right]=L[0.1 t] \\ L\left[T^{\prime}(t)\right]=0.1 L[t] \ldots \ldots(3) \end{array}\)

Consider the following Laplace transformation:

\(\begin{array}{l}L\left[T^{\prime}(t)\right]=T^{\prime}(s)\ldots\ldots(4)\\ L[t]=\frac{1}{s^2}\ldots\ldots(5)\end{array}\)

Substitute equations (4) and (5) in (3).

\(\begin{array}{l} T^{\prime}(s)=0.1 \frac{1}{s^{2}} \\ T^{\prime}(s)=\frac{0.1}{s^{2}} \ldots . .(6) \end{array}\)

Rearrange equation (1).

\(T_{m}^{\prime}(s)=\frac{1}{\left(\tau_{1} s+1\right)\left(\tau_{2} s+1\right)} T^{\prime}(s) \ldots\ldots(7)\)

Substitute equation (6) in (7).

\(T_m^{\prime}(s)=\frac{1}{\left(\tau_1s+1\right)\left(\tau_2s+1\right)}\frac{0.1}{s^2}\ldots\ldots(8)\)

Substitute 1 for \(\tau_{1}\) and 0.1 for \(\tau_{2}\) in equation (8).

\(\begin{array}{l} T_{m}^{\prime}(s)=\frac{1}{(s+1)(0.1 s+1)} \frac{0.1}{s^{2}} \\ T_{m}^{\prime}(s)=\frac{1}{0.1(s+1)(s+1 / 0.1)} \frac{0.1}{s^{2}} \\ T_{m}^{\prime}(s)=\frac{1}{s^{2}(s+1)(s+10)}\ldots\ldots(9) \end{array}\)

Apply partial fraction expansion for equation (8).

\(T_{m}^{\prime}(s)=\frac{\alpha_{1}}{s}+\frac{\alpha_{2}}{s^{2}}+\frac{\alpha_{3}}{(s+1)}+\frac{\alpha_{4}}{(s+10)} \ldots \ldots(10)\)

Substitute equation (9) in (10).

\(\begin{array}{l} \frac{1}{s^{2}(s+1)(s+10)}=\frac{\alpha_{1}}{s}+\frac{\alpha_{2}}{s^{2}}+\frac{\alpha_{3}}{(s+1)}+\frac{\alpha_{4}}{(s+10)} \\ \frac{1}{s^{2}(s+1)(s+10)}=\frac{\left[\begin{array}{c} \alpha_{1} s(s+1)(s+10)+\alpha_{2}(s+1)(s+10)+ \\ \alpha_{3} s^{2}(s+10)+\alpha_{4} s^{2}(s+1) \end{array}\right]}{s^{2}(s+1)(s+10)} \\ 1=\left[\begin{array}{c} \alpha_{1} s(s+1)(s+10)+\alpha_{2}(s+1)(s+10)+ \\ \alpha_{3} s^{2}(s+10)+\alpha_{4} s^{2}(s+1) \end{array}\right] \\ 1=\left[\begin{array}{c} \alpha_{1} s\left(s^{2}+11 s+10\right)+\alpha_{2}\left(s^{2}+11 s+10\right) \\ +\alpha_{3} s^{3}+\alpha_{3} 10 s^{2}+\alpha_{4} s^{3}+\alpha_{4} s^{2} \end{array}\right] \\ 1=\left[\begin{array}{c} \alpha_{1} s^{3}+\alpha_{1} 11 s^{2}+\alpha_{1} 10 s+\alpha_{2} s^{2}+\alpha_{2} 11 s \\ +\alpha_{2} 10+\alpha_{3} s^{3}+\alpha_{3} 10 s^{2}+\alpha_{4} s^{3}+\alpha_{4} s^{2} \end{array}\right]\ldots \ldots(11) \end{array}\)