Solution Found!
A perfectly stirred tank is used to heat a flowing liquid.
Chapter 14, Problem 14.4(choose chapter or problem)
A perfectly stirred tank is used to heat a flowing liquid. The dynamics of the system have been determined to be as shown in Fig. E14.4.
where:
\(P\) is the power applied to the heater
\(Q\) is the heating rate of the system
\(T\) is the actual temperature in the tank
\(T_m\) is the measured temperature
A test has been made with \({P}^{\prime}\) varied sinusoidally as
\({P}^{\prime} = 0.5\ sin\ 0.2t\)
For these conditions, the measured temperature is
\(T_{m}^{\prime}=3.464 \sin (0.2 t+\phi)\)
Find a value for the maximum error bound between \(T^{\prime}\) and \(T_{m}^{\prime}\) if the sinusoidal input has been applied for a long time.
Questions & Answers
QUESTION:
A perfectly stirred tank is used to heat a flowing liquid. The dynamics of the system have been determined to be as shown in Fig. E14.4.
where:
\(P\) is the power applied to the heater
\(Q\) is the heating rate of the system
\(T\) is the actual temperature in the tank
\(T_m\) is the measured temperature
A test has been made with \({P}^{\prime}\) varied sinusoidally as
\({P}^{\prime} = 0.5\ sin\ 0.2t\)
For these conditions, the measured temperature is
\(T_{m}^{\prime}=3.464 \sin (0.2 t+\phi)\)
Find a value for the maximum error bound between \(T^{\prime}\) and \(T_{m}^{\prime}\) if the sinusoidal input has been applied for a long time.
ANSWER:Step 1 of 5
Refer Figure E14.4 in the textbook for the dynamics of the system.
Thermocouple is used to measure the temperature in an air stream. It shows maximum and minimum temperature and varies at regular intervals.
From Figure E14.4, the transfer function of thermocouple is,
\(\frac{T_{m}^{\prime}(s)}{T^{\prime}(s)}=\frac{1}{0.2 s+1} \ldots \ldots \ldots(1)\)
Here,
Measured temperature is denoted as \(T^{\prime}{ }_{m}\)
Actual temperature is denoted as \(T^{\prime}\)
Rearrange equation (1) to find the actual temperature \(T^{\prime}\).
\(T^{\prime}(s)=T_{m}^{\prime}(s)(0.2 s+1) \dots \dots(2)\)