A first-order system is characterized by a time constant of τ = 0.1 s. The system is initially at a temperature of T0 = 30^◦C. It is subject to a periodic exchange of heat at a circular frequency of 33 rad/s with an ambient that is given by T∞ = 50 + 20 cos(33t) where t is in s. Determine the time

Question

A first-order system is characterized by a time constant of [latex]τ = 0.1 s[/latex]. The system is initially at a temperature of [latex]T_{0} = 30°C[/latex]. It is subject to a periodic exchange of heat at a circular frequency of 33 rad/s with an ambient that is given by [latex]T_{∞} = 50 + 20 cos(33t)[/latex] where t is in s. Determine the time lag between the input and the system response when the steady state has been reached. Make a plot showing both the input and the response in the same plot.

Accepted Answer

The problem will make use of the solution obtained above for a first-order system subject to periodic input.

1. The pertinent parameters in this problem are

Time constant: [latex]τ = 0.1 s[/latex]
Input wave circular frequency: [latex]ω = 33 rad/s[/latex]
Initial temperature of the system: [latex]T_{0} = 30°C[/latex]
Mean temperature of the input: [latex]T_{m} = 50°C[/latex]
Amplitude of the input disturbance: [latex]T_{a }= 20°C[/latex]

2. The phase lag is calculated referring to Fig. 3.4 as
[latex]δ = tan^{-1}(ωτ ) = tan^{-1}(33 × 0.1) = tan^{-1} 3.3 = 1.277 rad or 73.14°[/latex]

3. The time lag [latex]t_{L}[/latex] is defined through the relation [latex]δ = ωt_{L}[/latex] and hence we have

[latex]t_{L} =\frac{\delta }{\omega } =\frac{1.277}{33}=0.039s[/latex]

4. The amplitude reduction factor ARF is given by

[latex]ARF=\frac{1}{\sqrt{1+\omega^{2} au^{2} } } =\frac{1}{\sqrt{1+33^{2} imes 0.1^{2} } } =0.290[/latex]

5. Having completed the calculations, we make a plot as required in the statement of the problem. Since the mean temperature of the input is not zero, we need to combine (or superpose) the solutions obtained for the case of step input (Eq. 3.7) and the solution obtained for periodic input (Eq. 3.13). The solution turns out to be

[latex]T=\underbrace{\left[T_{0}-\frac{T_{a} }{1+\omega^{2} au^{2} } ight]e^{-\frac{t}{ au } } }_{Transient response} + \underbrace{\frac{T_{a} }{\sqrt{1+\omega^{2} au^{2}} \cos (\omega t-\delta ) }}_{Steady state response}[/latex] (3.13)

[latex]\underbrace{\frac{T-T_{\infty } }{T_{0} -T_{\infty }}}_{ heta } =e^{-\frac{t}{ au } }[/latex] (3.7)

[latex]T=T_{m} +\left[T_{0}-T_{m}-\frac{T_{a}}{1+\omega^{2} au^{2} } ight]e^{-\frac{t}{ au } } + \frac{T_{a}}{\sqrt{1+\omega^{2} au^{2}} } \cos (\omega t-\delta )[/latex]

[latex]or T=50-21.57e^{-\frac{t}{0.1} } +5.8\cos(33t-1.276)°C[/latex]

We have made a plot of this in Fig. 3.6. Time lag [latex]t_{L} ≈ 0.04 s[/latex] is also shown in the figure.

Question 3.3: A first-order system is characterized by a time constant of ...

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