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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.