6.3 Process Modelling
6.3.2 Model of the Steam-Drum Unit
As stated earlier, the drum is a nonlinear system with a strong coupling between its input and output channels in addition to a non-minimum phase response associated with the shrink and swell of the steam bubbles under the water level. Over the years, several models were introduced to capture the dynamical behaviour of the process, see for example [52, 77, 78]. In particular, we consider the so-called
˚Astr¨om - Bell model introduced in [73]; the model is the result of various improvements through the course of its development cycle which led to a 4-th order system with three actuating variables and two output channels capable of capturing most of the complicated dynamics occurring within the drum. The model basically considers mass-flow and energy balance at different parts of the drum: for the whole system, within the naturally-circulated downcomer-riser loop, and finally with regards to the condensation inside the drum, see Fig. 6.5. Shortly after, we introduce the governing mathematical equations describing dynamics of the systems. The model constant parameters are listed in Table6.1.
1The low-pressure steam-drum unit was previously regulated via the industry standard 3-element PID-controller. For further details with regards to this control architecture, the reader is referred to our previous work [41, Ch. 2].
Feedwater valve Economizer
Feedwater Tank
Feedwater pump
Superheater Valve Steam outflow
Evaporator Downcomer-Riser loop
Exhaust Heat from the gas Turbine Steam drum
Figure 6.4: Simplified illustration of the steam generation process. The red line (dotted) indicates hot steam and the blue line (solid) indicates cold water. The cold water inside the feedwater tank is pumped and heated at the economizer stage before going through the drum inlet. Due to the gravitational force, feedwater flows through the naturally circulated downcomer riser loop, where it is converted into steam at the evaporator stage. Different riser tubes collect the steam and supply it back into the drum. In the final stage, the saturated steam is taken from the drum outlet to the superheater.
6.3.2.1 Global Mass-flow and Energy Balance
The model is based on the assumption that most of the system parts will be under thermal equilibrium due to their direct contact with the saturated liquid/vapour mixture; in other words, the energy stored in the mixture is either absorbed or released quickly following changes in the drum pressure. This leads various metal parts of the system to adapt their temperatures in the same manner. This basic assumption agrees with experimental observation as it was proven that the temperature difference is almost negligible;
thus a detailed representation of the temperature distribution within the metal is not necessary.
With this basic assumption, the mass-flow and energy balance for the overall steam-drum unit are ex-pressed by:
qf w−qs= d dt
ρsVst+ρwVwt
, (6.4)
Q+qf whf w−qshs= d dt
ρshsVst+ρwhwVwt−P Vwt+mtCptsat
, (6.5)
with (6.4) and (6.5) describing to the balance of the mass-flow and the energy-flow, correspondingly. Here Vwt[m3] is the water total volume, whereas Vst[m3] is the steam total volume, Q[W] is the heat-flow rate associated with the gas turbine exhaust temperature,qf w[kg/s] andqs[kg/s] denote mass-flow rates of cold feedwater and superheated steam, respectively, andP[Pa] describes the drum absolute pressure.
Additionally, the thermal propertiesh[J/kg] andρ[m3/kg] describe the specific enthalpy and density at the saturation pressureP, correspondingly. Note that the subscripts s,w, t,f w refer to steam, water, total, and feedwater, respectively.
6.3.2.2 Mass-flow and Energy Balance of Downcomer-Riser Loop
Now we consider the mass-flow and energy balance for the naturally circulated loop with the evaporator.
Here, the governing differential equations are:
qdc−qr= d with αr[−] and αv[−] as the quality and average volume fraction of the steam within the riser tubes, correspondingly, hc :=hs−hw[J/kg] is the condensation specific enthalpy, andqdc[kg/s] and qr[kg/s]
corresponds to the mass-flow rate through the downcomer and riser tubes, respectively. The balancing equations (6.6) and (6.7) are based on a lumped model approximating the dynamics of water and steam inside a heated tube, governed by a complicated set of partial differential equations. Using this lumped model, one can expressqdcempirically via the following the algebraic equation [73]:
qdc=
6.3.2.3 Distribution of Steam inside the Drum
The final set of equations considers the mass-flow and energy balance through the water level inside the drum. Generally, it is extremely hard to develop a mathematical model from first principles with a reasonable degree of complexity considering the complication of the physical phenomena occurring inside the drum; thus, an empirical equation resulting from various attempts to fit with the experimental data was proposed in [73]. This empirical equation is expressed via:
d
Here the mass balance of the steam bubbles under the water level is defined in terms of the condensation flowqcd, in addition to the steam flow through the liquid surfaceqsd. This flow is driven by the density difference of the mixture, in addition to the momentum of the flowqr entering through the riser tubes.
In fact, many of the complex phenomena inside the drum can be captured by (6.16) using proper pa-rameterizations of the constants listed in Table6.1. Since we accounted for the distribution of the steam bubbles under the water level, we can now describe the level via a linearized behaviour expressed with the knowledge of the drum surface cross sectional areaAdl[m2]; that is
ld= Vwd−Vsd
Adl
:=lwd+lsd, with:Vwd=Vwt−Vdc−(1−αv)Vr
(6.10)
with lwd[m] and lsd[m] denoting variations of the level resulting from changes in the water and steam, respectively.
Feedwater inflow
Steam outflow
Downcomer inflow
Riser tubes inflow
Risers Condensation
outflow
Steam outflow through water
Evaporator drum
unit
Heat flow rate
Figure 6.5: Schematic diagram of the downcomer-riser circulation loop.
6.3.2.4 ODE Model
Combining all previous results, we can formulate a nonlinear model expressed via a set of explicit ODEs.
First we introduce the vectorxd:= P, Vwt, αr, VsdT
to describe the state variables of the drum. Here the drum pressureP is obviously chosen being a state since it describes the total energy of the system, the accumulation of water related to the total water volume Vwt in the system is selected being a state since it represents the storage of mass, whereas the steam quality αr in the riser tubes and the volume of the steam bubbles under the liquid level Vsd are chosen as state variables to describe distribution of steam under the water, thus making it possible to estimate the level as shown in (6.10). Additionally, we introduce the vectorsud:= Q, qf w, qs
T
andy:= P, ld
T
being the vectors of the input variables and the measurable outputs.
After definingVt:=Vwt+Vst and calculating the time derivatives in (6.4) and (6.5) via the chain rule, it is not hard to see that the mass- and energy-flow can be rewritten as
e11dVwt
with the variable coefficientsenm expressed via:
e11=Vwt re-sulted from the derivation of the density and enthalpy with respect to time and pressure. This is based on the fact their corresponding values, obtained from steam tables, change depending on the saturation pressure, which in turn varies over time.
In order to describe (6.6) and (6.7) as an ODE similarly to (6.11), we first eliminateqrfrom (6.6) and (6.7) by multiplying (6.7) by the factor (hw+αrhc), then adding the result to (6.6), and finally we evaluate the time-derivative via the chain rule. Thus the distribution of steam within the naturally circulated loop becomes:
e31dP
dt +e33dαr
dt =Q−αrqdchc, (6.13)
whose variable coefficientsenmare expressed by:
e31=
After tedious, but straight forward calculations, we can express (6.9) in terms of the drum pressure and
the steam quality inside the riser tube according to:
with the variablesenmdescribed by:
e41=Vsd∂ρs
Finally with simple arrangements of (6.11), (6.13) and (6.16), the drum model can be expressed via the following set of ODEs:
As mentioned earlier, the design and implementation of the controller to regulate the water level and the pressure inside the drum was part of a previously published work, see [45, 46]; hence we only briefly discuss the overall methodology and governing mathematical model.
Table 6.1: Parameters of the steam-drum model
Variable Description Unit
Vd Drum volume [m3]
Vr Riser volume [m3]
Vdc Downcomer volume [m3]
Vsd◦ Hypothetical volume of the steam [m3] τd Residence time of steam inside drum [s]
βd Empirical coefficient in (6.9) [−]
Fd Friction coefficient in downcomer-riser loop [−]
Cp Metal specific heat capacity [J/kg K]
tsat Saturation temperature ◦C
Adl Surface area of drum Water level [m2] Adc Downcomer cross-sectional area [m2]