Reference ID: MET-BB8F | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Proportional-only control is the simplest feedback strategy used in the process industries. It produces an immediate, linear correction to the final control element (valve, heater, pump, etc.) that is proportional to the instantaneous deviation between the measured variable and the set-point. Because the controller output is zero when the error is zero, a proportional controller must accept a permanent residual error—called offset—to generate the exact valve position required by the steady-state energy (or mass) balance. The calculation on this sheet quantifies that offset for a steam-heated juice pasteuriser, but the same approach applies to any self-regulating, non-integrating process where proportional control is acceptable and tight set-point tracking is not critical.
Methodology & Formulas
Energy balance on the process fluid
The steady-state heat load \(Q\) needed to raise the juice mass flow rate \(\dot{m}_{\text{juice}}\) from inlet temperature \(T_{\text{in}}\) to set-point temperature \(T_{\text{set}}\) is
\[
Q = \frac{\dot{m}_{\text{juice}}\;c_{p,\text{juice}}\;(T_{\text{set}}-T_{\text{in}})}{3600}
\]
where \(c_{p,\text{juice}}\) is the specific heat capacity (kJ kg\(^{-1}\) K\(^{-1}\)) and the divisor 3600 converts kJ h\(^{-1}\) to kW.
Required steam flow
Latent heat supplied by condensing steam at rate \(\dot{m}_{\text{steam}}\) must match the heat load:
\[
\dot{m}_{\text{steam}} = \frac{Q}{h_{\text{fg}}}\;3600
\]
with \(h_{\text{fg}}\) the latent heat of vaporisation (kJ kg\(^{-1}\)).
Valve position
The control valve is assumed linear and is sized for a maximum flow \(\dot{m}_{\text{max}}\). The demanded valve position \(m_{\text{valve}}\) (%) is
\[
m_{\text{valve}} = \frac{\dot{m}_{\text{steam}}}{\dot{m}_{\text{max}}}\;100
\]
Proportional controller equation
A proportional controller with gain \(K\) (dimensionless) and bias \(M\) (%) obeys
\[
m_{\text{valve}} = K\,e + M
\]
where \(e\) is the error expressed in percent of the transmitter span. Rearranging gives the steady-state error
\[
e_{\%} = \frac{m_{\text{valve}}-M}{K}
\]
Error in engineering units
If the temperature transmitter has span \(\Delta T_{\text{span}}\) (°C), the offset in degrees Celsius is
\[
e_{^\circ\mathrm{C}} = e_{\%}\;\frac{\Delta T_{\text{span}}}{100}
\]
Steady-state controlled temperature and offset
The juice temperature that the loop will settle at is
\[
T_{\text{ss}} = T_{\text{set}} - e_{^\circ\mathrm{C}}
\]
and the resulting offset is
\[
\text{offset} = T_{\text{set}} - T_{\text{ss}} \equiv e_{^\circ\mathrm{C}}
\]
Typical validity ranges for food-plant proportional controllers
Parameter
Lower limit
Upper limit
Remark
Controller gain \(K\)
0.2
2.0
Below 0.2 the loop is too sluggish; above 2.0 oscillations or instability appear.
Valve position \(m_{\text{valve}}\)
0 %
100 %
Negative or >100 % indicates valve saturation and the offset formula is no longer valid.
Calculated \(T_{\text{ss}}\)
0 °C
—
Negative temperatures are physically impossible for aqueous food streams.
A proportional controller multiplies the present error by the gain Kc; it never “pushes” the valve far enough to eliminate the error completely. The steady-state offset (also called droop) is calculated from the process gain Kp and the controller gain Kc:
Offset = ΔSP / (1 + Kc·Kp)
where ΔSP is the sustained set-point change. The larger the product Kc·Kp, the smaller the offset, but it can reach zero only if Kc is infinite—which is impossible in real life.
Treat the load change as an input disturbance and use the same steady-state balance. The resulting offset is:
Offset = ΔLoad / (Kc·Kp)
Note that the disturbance must be expressed in the same engineering units as the controlled variable. If you need zero steady-state error for load upsets, add integral action (PI or PID).
Raising Kc reduces offset, but the loop will eventually oscillate or go unstable. The maximum usable gain is set by:
Process dead-time and lag (use the Ziegler–Nichols ultimate-gain test)
Valve saturation and mechanical wear
Noise amplification that causes excessive valve travel
Always verify the gain margin (≥ 2) and phase margin (≥ 30°) before accepting a higher Kc.
Integrating processes have an inherent “memory,” so a pure P-controller can achieve zero offset for set-point changes. However, load disturbances still produce a steady-state error given by:
Offset = ΔLoad / Kc
Because the process gain Kp is effectively infinite at steady state, the 1 + Kc·Kp term collapses to Kc·Kp, cancelling Kp and leaving only Kc in the denominator.
Worked Example – Proportional-Control Offset in a Juice Heater
A process engineer is commissioning a shell-and-tube heater that uses 3.9 kJ kg⁻¹ °C⁻¹ clarified apple juice as the cold stream and 2200 kJ kg⁻¹ latent heat steam as the hot utility. The juice must leave the exchanger at 90 °C. A proportional-only controller throttles the steam valve; the valve is sized for 400 kg h⁻¹ maximum flow and the transmitter span is 50 °C. With the plant running at 2000 kg h⁻¹ juice flow the operator notes a steady-state temperature below set-point. Determine the inherent proportional offset.
Knowns
Controller gain \(K = 0.8\) %/%
Juice specific heat capacity \(c_{p,\ juice} = 3.9\) kJ kg⁻¹ °C⁻¹
