3.2 The ‘state detrmined’ library
3.2.4 The hot drum
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Fig. 3.28. Organic fluid vapour fraction measured at varying distance from pipe inlet by Takamatsu et. al.
[32] (dots) and calculated from the model with different values of damping coefficient.
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89 3.2.4.1 Mathematical model
The following equations are considered:
Mass balance:
The difference in the mass flow rates entering and leaving the drum must result in a mass change in time. Expressing this principle with finite difference approach, the mass stored within the component can be calculated in the next step of simulation t+∆t:
( )
t t t t t t t t
l v l v in out
m +∆ +m +∆ =m +m + ∆t m −m (3.57)
where the total mass is expressed as sum of the vapour and liquid masses.
Energy balance
The overall energy balances states that the energy flow entering the system with the organic fluid coming from the evaporator (or condenser), and the energy flow leaving the drum (that, if possible, is achieved by subtracting saturated vapour), must result in a system energy change in time. Expressing the energy conservation equation according to the finite difference approach, the internal energy at next step of simulation can be expressed through the following equation:
( )
t t t t t t t t t t t t t t t t
l l v v l l v v in in out out
m
+∆u
+∆+ m
+∆u
+∆= m u + m u + ∆ t m h − m h
(3.58)Volume conservation
Indeed another physical condition to be considered is that volume of the liquid phase and vapour phase must equal the overall drum internal volume, that is assumed constant in time (rigid walls):
t t t t
v l
t t t t TOT
v l
m m
V
+∆ +∆
+∆ + +∆ =
ρ ρ (3.59)
State equations
Since the component features a number of 7 state variables (ml, mv, p, ul, uv ρl and ρv), four equations are needed for system closure. These equations come from applying different times the fluid equation of state that is in fact used to determine:
( ) ( )
( ) ( )
, , 1
, , 0
t t t t t t
v v
t t t t t t
l l
u f p x
u f p x
+∆ +∆ +∆
+∆ +∆ +∆
= =
= =
ρ
ρ (3.60)
The Simulink® model of the hot drum component and its dialog window interface are shown in Fig.
3.30. To be noted that the parameters required are geometrical parameters and parameters required for initializing the system at simulation time t=0.
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(a) (b)
Fig. 3.30. Simulink® model of the drum (a) and block dialog mask (b).
The existence of state parameters implies that initialisation parameters should be provided so, the Simulink® block dialog musk must include, besides geometrical and physical parameters as drum size and fluid adopted, also the initial values of pressure, mass of liquid and mass of vapour at simulation time t=0.
To be noted that, among the different parameters to be provided to the model, the overall volume of the drum has significant effect on determining the time evolution of the main system outputs. Fig. 3.31 shows for example the effects played by drum volume in the dynamic response due to a step change in one of the system inputs; R123 is considered as organic fluid. It can be noted that, as expected, the smallest the drum volume, the faster is the system response.
(a) (b)
Fig. 3.31. Pressure response in the hot drum (b) due to a step change in the enthalpy flow of the fluid entering (a) at varying drum volumes.
3.2.4.2 Dynamic behaviour of the drum-evaporator subsystem
The mutual interaction between hot drum and evaporator are strong and the dynamic behaviour of each component cannot be really separated from the other. In fact what happens within the evaporator has
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91 effect on the evaporator outputs as for example the organic fluid mass flow rate or enthalpy, which in turn, affects the energy and mass conservation equations of the hot drum that would result in a new pressure within the component, that is fed backward to evaporator and so on.
Evaporator and hot drum have therefore been tested together and the following plots are proposed as example of their mutual interactions. The model of the evaporator linked to the model of the hot drum in the Simulink® environment is proposed in Fig. 3.32.
Fig. 3.32. Simulink® model of an evaporator-hot drum system.
The system has been operated using the following geometry of the evaporator:
L=95m;
d=45·10-3m;
di=50·10-3m;
do=85·10-3m.
The organic fluid employed is R123 and the transfer fluid is diathermic oil. The drum volume is assumed to be 0.75m3, while the number of nodes adopted for the axial discretization of the evaporator is n=20. The number of nodes has been chosen as a result of a convergence analysis.
In Tab. 3.2 the value of pressure existing within the evaporator-drum system at varying number of nodes is reported. It can be observed that the higher the number of axial nodes considered the lower is the system pressure. This is a consequence of the fact that the organic fluid specific enthalpy entering the drum is lower when a low number of nodes is considered since it is evaluated as average over a bigger control volume (the length of each pipe cv is bigger). It can also be noticed that increasing from 20 to 30 nodes the pressure change is negligible and usually, when smaller than 5%, it can be assumed that convergence is reached.
n Pvap [kPa]
(steady state)
Relative variation [%]
5 1586 -
10 1885 +18.85
20 2051 +8.81
30 2104 +.258
Tab. 3.2. Analysis of convergence on the number of nodes of the axial discretization of the evaporator.
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After reaching steady state operating conditions one of the main system external inputs is changed with a step. In this example the temperature of the heat transfer fluid is changed and this has effects on both the hot drum (Fig. 3.35) and the evaporator (Fig. 3.34).
It can be observed that the decreasing transfer fluid temperature determines a decrease in the pipe wall temperature (that is sharper close to the transfer fluid inlet, Fig. 3.35 (b) ) which in turn determine a decrease in the organic fluid specific enthalpy and temperature (Fig. 3.35 (d) and (e) ). The decreased energy flow that the evaporator provides through the organic fluid to the drum causes a decrease in the pressure existing within the system (Fig. 3.34 (a) ) which in turn causes a fraction of the liquid present within the component to evaporate (Fig. 3.34 (b)).
At the end of transient all values stabilizes into a new steady state; to be noted that the pressure and hence the saturation temperature within the evaporator are at a lower value. Interesting is also Fig. 3.35 (e) which shows the nodal organic fluid mass flow leaving each discretized cell within the evaporator. To be remembered that the organic fluid mass flow rate entering the evaporator does not change with time.
The sudden drop in the wall temperature make the vapour within some cells to condensate thus reducing the actual mass flow rate through these cells, resulting in an overall decrease in the organic fluid mass flowrate in the two phase region. The subsequent fall in the evaporating pressure causes start of vaporization in some cell were, at the previous pressure level, liquid phase still existed. Part of the mass contained within these cells is then discharged, causing an increase in the leaving mass flow rate in all the following cells. To be noted that when steady state conditions are reached the mass flow rate flowing through each cell remains constant. Also it can be appreciated that no changes in the mass flow rate can be observed, even during transients, in the zone where only liquid exist (whose density is independent from existing pressure and temperature within small changes of these parameters).
Fig. 3.33. Change in temperature of the transfer fluid entering the evaporator.
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93
(a) (b)
Fig. 3.34. Hot drum response: pressure (a) and liquid volume fraction (b).
(a) (b)
(c) (d)
Fig. 3.35. Evaporator response: (a) transfer fluid temperature, (b) pipe wall temperature, (c) organic fluid temperature and (d) organic fluid mass flow rate. Distribution with time and as function evaporator abscissa.