3.2 The ‘state detrmined’ library
3.2.3 Dynamic model of a counterflow heat exchanger with phase change
3.2.3.1 Heat flux correlations in the internal pipe heat exchange with evaporating-
Under normal operating conditions the organic fluid enters the heat exchanger as subcooled liquid and leaves the exchanger as saturated or superheated vapour if the component operates as evaporator, conversely it may enter as satured or superheated vapour and live as liquid if the component operates as condenser. For this reason different fluid phase regions may exist within the ith cell of the internal pipe of the evaporator and different heat exchange correlations apply depending on whether the fluid is single phase (hi<hl or hi>hv) or two phase (hl<hi<hv).
In case of single phase the approach is similar to that adopted for the anular pipe, where the fluid is always assumed to be single phase, and Nu=4.36 in the laminar region otherwise the Gnielinski correlation (3.33) is applied.
In the two-phase fluid region the Chen correlation is instead adopted for determining the Nusselt number [28,29].
Dc Db
Nu=Nu +Nu (3.44)
NuDc expresses the Nusselt number for vaporization in forced convection, written from the Dittus-Boelter correlation [30]:
0.8 0.4
0.023 Re Pr
Dc l l
Nu = F (3.45)
where F is a correction empirical factor that takes into account of the characteristics of the fluid flow and calculated from the Martinelli factor Xtt [30]:
....
( ) ( ) ( )
,0 tf
tf
tf
m t T t p t
( )
( ) ( ) ( )
,1
,1
,1
,1
tf n
p n
f n
f n
T t T t h t ρ t
( )
,1
Ttf t
( )
,
hf n t
( ) ( ) ( )
,0 f
f
f
m t h t p t
mf n,
( )
tinputs state variables outputs
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82
0.5 0.1
1 0.9
Xtt v l
l v
x x
−
=
ρ µ
ρ µ (3.46)
and:
0.736
2.35 1 0.213 if X 10
1 if X 10
tt tt
tt
F X
F
= + <
= >
(3.47)
The Reynolds number for the liquid can be calculated as:
( )
4 1 Rel
l
m x
d
= −
π µ (3.48)
NuDb represents instead the Nusselt number in case of nucleation boiling, and can be expressed according to the following:
0.24 0.5 2 0.5
0.24 0.21
0.00122 Pr l l 2
Db l
v l
d p
Nu S Ja d p∆ ∆
=
ρ ρ
ρ σ µ (3.49)
In the previous correlation S is a factor smaller than 1 defined by the following empirical correlation:
6 1.17
2
1 1 2.53 10 Re F
S= −
+ ⋅ (3.50)
and takes into account of the decreasing importance of nucleation with increasing two phase Reynolds number Re2F, defined as:
1.25
Re2F=RelF (3.51)
The Jacob number in Eq.(3.49) can be defined as:
l e
lv
Ja c T H
= ∆ (3.52)
where ∆Te=Tp-Tvs is the difference between pipe wall temperature and fluid saturation temperature, and Hlv is the latent heat of vaporization.
The term ∆p of equation (3.49) expresses the difference between vapour saturation pressure at the wall temperature and the liquid hydrostatic pressure:
ALIBRARY OF MODELS FOR THE DYNAMIC SIMULATION OF ENERGY SYSTEMS
83
( )
vs p l
p p T p
∆ = − (3.53)
Once the Nusselt number is calculated from Eq.(3.44), the convection coefficient can finally be defined as:
Nu l
h= dλ
(3.54)
and the actual overall convective thermal flow exchanged between pipe and evaporating organic fluid, regardless if in the single phase or two phase region, can be expressed as:
( )
,
conv i p f
q =h T −T
π
d x∆ (3.55)To model the heat exchange between pipe and organic fluid some issue related to discontinuities in the functions defining the heat exchange coefficients must be taken into account. It should be noted in fact that when the organic fluid is single phase the fluid exchanges sensible heat with the pipe wall, while when phase change occurs the latent heat is involved, which is usually on order of magnitude higher than sensible heat [27]. This implies that, due to the discretized approach adopted, in neighbouring organic fluid control volumes the heat transfer coefficient may experience a step change if the fluid is first subcooled (node i) then it starts to vaporize (node i+1) or, equally, it is ending the vaporization process (node i) and in the following node (i+1) it is completely in the vapour field. These discontinuities in the correlations adopted cause serious stability problems to the solution of the explicit set of equations presented in this section. For this reason a “dumping” coefficient is introduced in order to make smoother the change in the heat transfer coefficient and, while representing an approximation, it turns out to be extremely useful in reducing stability concerns of the model and rising computational speed. It must also be considered that, as stated in [29], the validity of Chen correlation for x>0.85 is limited, hence the approximation can be accepted. It will be also shown that the effect of approximating the step change in the convection coefficient with a ramp is negligible.
Fig. 3.24 reports the heat transfer coefficient as function of the vapour fraction x in the two phase region as from the Chen correlation and in the hypothesis of smoothing the step change for x=0 and x=1.
The “dumping coefficient”, dc, is assumed to be the ∆x, from x=0 and x=1, in which the two phase convection coefficient is approximated as a straight line linking the convection coefficient calculated from Chen at x= ∆x and x=1-∆x and the convection coefficient for the liquid at x=0 and for the vapour at x=1. In Fig. 3.24 for example the “dumped” convection coefficient is calculated with dc=0.1 and, in fact, the Chen correlation is adopted in the range 0.1<x<0.9, otherwise two straight lines approximate the convection coefficient linking the values in a smoother way to the liquid and vapour coefficients. The example refers to a pipe with d=0.02m where a mass flow rate of 0.1kg/s of water at atmospheric pressure vaporizes due to the pipe wall temperature assumed constantly at 120°C.
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84
Fig. 3.24. Heat transfer coefficient in the two phase zone as function of vapour fraction.
Before proceeding further with illustrating the complete Simulink® model of the evaporator some of the strongest assumptions introduced in the model have also to be verified.
One of these is the hypothesis of lumped thermal capacitance for the metal that has brought to assuming the temperature of each pipe element related to any discretized volume to be at constant and uniform temperature. As shown in Par.3.2.1.2 this hypothesis can be accepted as far as Bi<<0.1. In the case under analysis the condition Bi<<0.1 is verified both with reference to the transfer fluid – pipe and with organic fluid – pipe heat transfer convection coefficient, hence the assumption of lumped thermal capacitance for the metal pipe can be accepted.
The second strong hypothesis introduced in the analysis is that of constant pressure within the pipe where the organic fluid flows. Besides the pressure losses due to friction between the flowing fluid and the pipe which, within some extent can reasonably be neglected, another important phenomena occurs when a fluid that undergoes phase change flows within a straight pipe, that has been called here
“backpressure of inertia” [31].
It is well known that, for a fluid that undergoes phase change, as for the organic fluid flowing within the evaporator inner pipe, the fluid density changes within the flow stream both because of friction and because of heat exchange processes. Applying Bernulli equation to the straight and constant cross section circular pipe, and neglecting the effects of friction, a decrease in the fluid density corresponds to a decrease in the fluid hydrostatic pressure being conversely increased the fluid speed and hence its volumetric flow ratio. For this reasons, also in the hypothesis of completely neglecting friction, a pressure drop along the pipe should still be considered.
The backpressure of inertia can be calculated according to the following equation from the inlet to the outlet section of the evaporator, with reference to the evaporating fluid:
2 ,
, , , ,
, f in 1
f in f out f in f in
f out
p p p c
∆ = − = −
ρ ρ
ρ (3.56)
ALIBRARY OF MODELS FOR THE DYNAMIC SIMULATION OF ENERGY SYSTEMS
85 For all the geometries considered in the present study the fluid speed at the evaporator inlet section, cf,in has always been limited to values smaller than 3m/s and the corresponding backpressure of inertia turned out to be limited to at most 6% of the pressure at the inlet section, pf,in: its effect have been neglected for sake of simplicity. In the future model upgrades will however consider the phenomena by introducing the momentum conservation equation.