Reference ID: MET-3D87 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
A first-order system response describes how a temperature probe (or any thermal mass) follows a sudden change in the surrounding fluid temperature. In process engineering this is critical for selecting sensors, tuning control loops, and estimating measurement lags in heat-exchangers, reactors, and pipelines. The lumped-capacitance model—valid when internal temperature gradients are negligible—reduces the energy balance to a single exponential lag characterised by a time constant \( \tau \).
Methodology & Formulas
Geometry
Volume of cylindrical probe
\[ V = \pi \left( \frac{D}{2} \right)^2 L \]
Total heat-transfer area (lateral plus both ends)
\[ A = 2\pi \left( \frac{D}{2} \right) L + 2\pi \left( \frac{D}{2} \right)^2 \]
Characteristic length for conduction
\[ L_c = \frac{V}{A} \]
Biot number
Ratio of internal conduction resistance to external convection resistance
\[ \mathrm{Bi} = \frac{h\,L_c}{k} \]
Regime
Criterion
Lumped capacitance valid
\(\mathrm{Bi} < 0.1\)
Spatial temperature gradients significant
\(\mathrm{Bi} \geq 0.1\)
Time constant
Product of thermal capacitance and external resistance
\[ \tau = \frac{\rho\,V\,c}{h\,A} \]
The probe temperature evolves as
\[ \frac{T(t)-T_{\infty}}{T_0-T_{\infty}} = \exp\left(-\frac{t}{\tau}\right) \]
where \( T_0 \) is the initial probe temperature and \( T_{\infty} \) is the new fluid temperature.
Reynolds number
\[ \mathrm{Re} = \frac{u\,D}{\nu} \]
Flow regime
Re range
Laminar
\(\mathrm{Re} < 2300\)
Transitional
\(2300 \leq \mathrm{Re} < 4000\)
Turbulent
\(\mathrm{Re} \geq 4000\)
The convection coefficient \( h \) is usually obtained from empirical correlations that depend on \( \mathrm{Re} \) and \( \mathrm{Pr} \); check the correlation limits before use.
The time constant τ is the time it takes the process variable to reach 63.2% of its total change after a step change in the manipulated variable.
Record the initial steady-state value PV₀ and the final steady-state value PV∞.
Determine 63.2% of the total change: ΔPV63 = PV₀ + 0.632 × (PV∞ − PV₀).
Read the time at which the PV crosses ΔPV63; subtract the step initiation time to obtain τ.
Steady-state gain Kp relates the change in process output to the change in controller output.
Measure the total change in process variable ΔPV = PV∞ − PV₀.
Record the magnitude of the step change in controller output ΔOP.
Calculate Kp = ΔPV / ΔOP (use engineering units for both variables).
Use the analytical solution for a first-order step response:
PV(t) = PV₀ + Kp·ΔOP·(1 − e−t/τ)
Insert the known values of Kp, ΔOP, and τ.
Evaluate the exponential term for the desired time t.
Add the result to the initial value PV₀ to obtain the predicted PV(t).
A first-order system is considered settled within approximately 4τ.
At t = 3τ the response has reached 95% of the total change.
Therefore, 95% settling time tₛ ≈ 3τ for most regulatory tuning guidelines.
Worked Example – Verifying the Response Time of a Thermowell in a Warm-Water Line
A process engineer needs to confirm that a stainless-steel thermowell installed in a 0.15 m s-1 water line will reach 98% of a 40 °C → 80 °C step change within the plant’s 60 s alarm window. The following calculation checks the first-order time constant and the resulting 98% response time.
Knowns
Density, ρ = 8000 kg m-3
Specific heat, c = 500 J kg-1 K-1
Thermal conductivity, k = 16 W m-1 K-1
Outside diameter, OD = 0.006 m
Immersion length, L = 0.05 m
Water velocity, u = 0.15 m s-1
Initial water & probe temp, Tfluid,initial = Tprobe,initial = 40 °C
Final water temp, Tfluid,final = 80 °C
Convection coefficient, h = 400 W m-2 K-1
Kinematic viscosity water, ν = 6.58 × 10-7 m2 s-1
Prandtl number water, Pr = 4.34
Step-by-step calculation
Compute the wetted surface area of the thermowell tip (treated as a hemisphere plus cylinder, but for brevity we use the supplied value):
\[ A = 0.000999 \; \text{m}^2 \]
Compute the wetted volume:
\[ V = 1.414 \times 10^{-6} \; \text{m}^3 \]
Evaluate the Biot number to verify lumped-capacity validity:
\[ Bi = \frac{h \, L_c}{k} = \frac{400 \times 0.001415}{16} = 0.035 \]
Because \( Bi < 0.1 \), the lumped model is acceptable.
Determine the first-order time constant:
\[ \tau = \frac{\rho \, c \, V}{h \, A} = \frac{8000 \times 500 \times 1.414 \times 10^{-6}}{400 \times 0.000999} = 14.151 \; \text{s} \]
The thermowell reaches 98% of the 40 °C step in 55.4 s, which is within the 60 s alarm window. The design is therefore acceptable for the intended service.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
