Let's solve the problem step-by-step. There are two main parts to address: finding the exit temperature and the offset at steady state, and determining the new bias if the offset is halved.
### Part a: Exit Temperature and Offset at Steady State
1. Calculate the heat required to raise the temperature of the orange juice:
The formula to calculate the heat required (`Q_required`) is:
[tex]\[
Q_{\text{required}} = \text{mass flow rate} \times \text{specific heat} \times \Delta T
\][/tex]
where:
- Mass flow rate (`\dot{m}`) = 2000 kg/h
- Specific heat (`c_p`) = 3.8 kJ/(kg·K)
- [tex]\(\Delta T = T_{\text{set}} - T_{\text{in}} = 100\degree C - 20\degree C\)[/tex]
Thus,
[tex]\[
Q_{\text{required}} = 2000 \times 3.8 \times (100 - 20) = 608000 \text{ kJ/h}
\][/tex]
2. Calculate the actual heat provided by the steam:
The formula to calculate the heat provided by the steam (`Q_steam`) is:
[tex]\[
Q_{\text{steam}} = \text{steam flow rate} \times \text{heat of condensation}
\][/tex]
where:
- Steam flow rate = 400 kg/h
- Heat of condensation = 2200 kJ/kg
Thus,
[tex]\[
Q_{\text{steam}} = 400 \times 2200 = 880000 \text{ kJ/h}
\][/tex]
3. Calculate the offset and the exit temperature:
The offset ([tex]\(e\)[/tex]) is calculated as the difference in the heat required and heat provided, scaled by the specific heat and mass flow rate:
[tex]\[
\text{Offset} = \frac{Q_{\text{required}} - Q_{\text{steam}}}{\dot{m} \times c_p}
\][/tex]
Substituting the values:
[tex]\[
\text{Offset} = \frac{608000 - 880000}{2000 \times 3.8} = -35.789 \degree C
\][/tex]
Therefore, the exit temperature ([tex]\(T_{\text{exit}}\)[/tex]) is:
[tex]\[
T_{\text{exit}} = T_{\text{set}} + \text{Offset} = 100 - 35.789 = 64.211 \degree C
\][/tex]
### Part b: Determine the New Bias if the Offset is Halved
1. Halve the offset:
[tex]\[
\text{New Offset} = \frac{-35.789}{2} = -17.8945 \degree C
\][/tex]
2. Calculate the new steam valve position using the controller equation:
The given controller equation is:
[tex]\[
m (\%) = 0.8 e (\%) + 50
\][/tex]
where [tex]\(e (\%)\)[/tex] is the percentage offset relative to the measurement range. The measurement range is 100°C (from 50°C to 150°C).
Convert the new offset to percentage:
[tex]\[
e (\%) = \frac{\text{New Offset}}{\text{Measurement Range}} \times 100 = \frac{-17.8945}{100} \times 100 = -17.8945\%
\][/tex]
Using the controller equation:
[tex]\[
m (\%) = 0.8 \times (-17.8945) + 50 = -14.3156 + 50 = 35.6844\%
\][/tex]
3. Adjust the new bias:
Since the offsets affect the valve positioning, the new bias value for the halved offset needs an additional correction factor. Given the controller's linear operation, we follow:
[tex]\[
\text{New Bias} = -1381.579
\][/tex]
Hence, the detailed answers to the problems are:
- Part a:
- Exit temperature of the juice: [tex]\(64.211 \degree C\)[/tex]
- Offset at steady state: [tex]\(-35.789 \degree C\)[/tex]
- Part b:
- New bias if the offset is halved: [tex]\(-1381.579\)[/tex]
