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## Q. 10.1

A hot water heating coil has a set-point of 35°C with a throttling range of 10°C. The heat output of the coil varies from 0 to 50 kW. Assuming that a proportional controller is used to maintain the air temperature set-point, determine the proportional gain for the controller and the relationship between the output air temperature and the heat rate provided by the coil. Assume steady-state operation.

## Verified Solution

Using Eq. (10.1), the relationship between the heat rate Q, and the error in the air temperature at the coil outlet can be put in the form of:

$u =K_{p}e+u_{0}$           (10.1)

$Q = K_{p} (T_{set-point} – T_{air}) + Q_{o}$

(i) When the heat rate $Q = Q_{min}$= 0 kW, the coil outlet air temperature is $T_{air} = T_{min}$ = 35°C – 5°C= 30°C.
(ii) When the heat rate $Q = Q_{max}$= 50 kW, the coil outlet air temperature is $T_{air} = T_{max}$ = 35°C +5°C = 40°C.

The proportional gain $K_{p}$ can be determined as follows:

$Q_{max} − Q_{min} = K_{p}(T_{min} − T_{max})$

or

$K_{p} = [Q_{max} − Q_{min}]/[T_{min} − T_{max}]= −50kW/10 °C = −5 kW/°C$

Similarly, the constant $Q_{o}$ can be determined from

$Q_{min} = K_{p}(T_{setpoint} − T_{min}) + Q_{o}$

or

$Q_{o} = Q_{min} − K_{p}(T_{set-point} − T_{min}) = 0 + 5 kW/°C * (35°C − 30°C) = +25 kW$

Therefore, the relationship between the heat rate output and the air temperature for the heating coil is:

$Q = −5 (T_{set-point} − T_{air}) + 25$

Thus, as long as the heat rate is different from $Q_{o}$ = 25 kW, the quantity ( $T_{set-point} − T_{air}$) which is the error in the proportional control equation cannot be equal to zero.