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# Question: a shower system is manually controlled by the person who...

###### Question details

A shower system is manually controlled by the person who is having a shower as shown in Figure 1. Figure 1 – Schematic of manually control shower system

Several system components are working together:

• The mixer mixes hot and cold water streams to one single stream.
• Hot and cold water supply
• Human sensing.
• Control of the mixer handle is through the brain driving the hand to turn an angle.

The mixer has an outgoing temperature T to angle $\theta$ relationship: $T={K}_{1}\theta +\frac{{K}_{2}}{\theta }$                                                    (1)

where:

• $\theta$ = angle turned by the motor (rad) (measured from vertical line on the mixer as shown in Figure 2)
• T = water temperature from the mixer (deg C)
• K1 = first order coefficient constant (deg C/rad)
• K2 = second order coefficient constant (deg C/rad2) Figure 2 – Mixer temperature change according to angle turned

The human sensing component follows a root square relationship:

${T}_{m}={K}_{m}\sqrt{T}$

where

• Km is a constant

The brain controls the hand with an error message $ϵ$ determined from a desirable water temperature value.  The error message drives the hand to turn the mixer handle with an angle $\theta$:

$ϵ={K}_{B}\left({T}_{d}-{T}_{m}\right)$

$\theta ={K}_{n}ϵ$

where:

• Kn is a constant
• KB is a constant
• Tm is the sensed temperature
• Td is the desirable temperature

The user’s desirable temperature range is 35oC ± 15oC.  The output temperature from the system should be as close to the corresponding set value of desirable temperature.  Minor deviation from the set value is acceptable.

Which system parameter set would you recommend?

Group of answer choices
 K1 5 K2 -30 Kn 2 KB 0.004 Km 3
 K1 6 K2 -200 Kn 2 KB 0.04 Km 9
 K1 -500 K2 300 Kn 0.0001 KB 0.004 Km 200
 K1 -2 K2 1 Kn 0.0001 KB 0.004 Km 200
 K1 -2 K2 1 Kn 0.01 KB 0.03 Km 0.03
 K1 6 K2 -100 Kn 2 KB 0.04 Km 4

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