CHAPTER .3 MODEL DESCRIPTlON
Boiling point elevation
A series of attempts has been made over the years to correlate the boiling point elevation of sucrose solutions with sucrose concentration. All of these studies were, however, based on a severely limited number of data points provided by the individual authors. Moreover, only a few were based on sound thennodynamic principles, utilising the concept of the activity coefficient. None of these correlations was thus considered to be ideally suited for use in the climbing film evaporator model Thus, in order to improve the reliability of boiling point elevation prediction, Starzak and Peacock (1997) collected boiling point temperature, vapour pressure, equilibrium relative humidity and other closely related data for aqueous sucrose solutions from 56 studies published over more than a century. These data were processed to derive a thermodynamically rigorous and statistically sound equation for the water activity coefficient. which can be used to predict the boiling point elevation (in °C) of sucrose solutions over the full :range of sucrose concentrations and at different system pressures:
CHAPTER 3 MODEL DESCRIPTlON
This boiling point elevation prediction correlation was compared with twenty other well known prediction methods available in the literature and shown to be the most accurate and reliable of all the methods tested, with a mean deviation of about 2% (Starzak and Peacock, 1998) .
Density of condensate
The density of condensate (in kg/dm') was calculated using the correlation of Aggarwal (1989) for the specific volume of saturated water:
9
+
"A
L... n tnc - 4 0=5where Vc is the specific volume of condensate. tc is the condensate temperature in degrees Kelvin and the coefficients AI through ~ are given by
For I, s 373,2 K
A, =0.31060619e+01; A, = -0.90267563e+01; A, = 0.51 853027e+02;
A"
= -0.39588600e+02;A, = 0.39760528e+01;
A"
= -0.5 I 528862e+02;A, = 0.12148573e+03; As = -0.14039282e+03;
A"
= 0.65221863e+02;For 373.2 K < I, S 600 K
A, = 0.31060000e+01;
A, = -0,56161909e+01; A, = 0.13398 111e+02;
A, = 0.58633842e+01;
CHAPTER 3
A, ~ -0.18599625e+02;
A.
~ 0.64106083e+01;A, ~ -0.87578869e+01; As ~ 0.88766804e+01; A, ~ -0.37156558e+01;
For 1< > 600K
A, ~ 0.31060000e+01;
A, ~ -0.54649353e+01;
A, ~ -0.72486047e+03;
A, ~ 0.12055044e+04;
A, ~ -0.57213086e+03;
A, ~ 0.72626416e+03; A, ~ -0.65027012e+04; As ~ 0.41431125e+05;
A, ~-0.11816734e+06;
Density of sugar cane juice
MODEL DESCRIPTION
The density of sugar cane juice (in kg/m') was calculated using the correlation of Kadlec el. al. (1983):
PL ~ a+bTL+cT~+dT{
where the coefficients a, b, c and d are given by:
a ::::;
ao + al Bx + a2 Bx2 + a3 Bx3, ,
b = bo + bl Bx + b, Bx + b, Bx
d = do + dl Bx + d, Bx' + d, Bx'
CHAFTEA3 MODEL DESCRIPTION
where Bx is the dry solids content (sucrose and impurities) of the sugar cane juice on a mass percent basis. and the coefficients are given in Table 1.
Table 1. Coefficients for the Kadlec et, ai_ density correlation_
Concentration range i- 1 2 3 4
a 1000,45 3,94325 0,0146409 2,69936 x 1 O-~
b -6,01137 x 10-3 -6,85707 x 10-) -2,63869 x 10-<5 -1,54649 x 104 Bx :s 69%
c -5,44367 x 10'3 7,64646 x 1O,s -6,50649 x 10-7 8,44748 x 10-9 d 1,31672 x lO-s -3,55879 X 10-' 6,36639 x 10-9 -7,25049 x 10-1\
a 1316,33 -Q,61119 0,130327 -3,91182 x 10-' b -1,7077 0,0299153 -1,46234 x 10'" -4,69390 X 10-7 Bx>69%
C 6,51225 x 10-3 -1,65477 x 1~ 2,15744 x 10'7 8,36737 x 10'"
d 0 0 0 0
Density of vapour and steam
The density of vapour was calculated using the ideal gas equation.
Enthalpy of saturated liquid water
The specific enthalpy of saturated liquid water (in kJ/kg) was calculated using the correlation of Aggarwal (1989):
9
+
"A
L.... n tDw - 40 ; 5
where
hw
is the specific enthalpy, t,.,.. is the temperature in degrees Kelvin and the coefficients AI through A, are given byFor '. $ 287.2 K
AI ~ 0.2364914ge-02;
A, ~ 0.10918589e+04;
A, ~ -0.92032929e+04;
CHAPTER 3
A.
= 0.22190031e+05;As = -O.78623057e+04;
A"
= -0.32543672e+05;A, = 0.40901742e+05;
A, = -0.36393298e+03;
A"
= -0. 19495543e+05;For t. > 287,2 K
A, = 0.20860000e+04;
A, = -0.10864822e+04;
A, = 0.30911332e+05;
A.
= -0.44055891e+05;As = 0.11541795e+05;
A"
= 0.48651314e+02;A, = -0. 18603667e+04;
A, = 0.24305122e+04;
A"
= -0. 13371470e+04;Enthalpy of saturated water vapour
MOOEL. OESCR1PT10N
The specific enthalpy of satuIated water vapour (in kJlkg) was calculated using the correlation of Aggarwal (1989):
9
+ ' " A tn- 4
L... a v
a.S
where
h.
is the specific enthalpy,t.
is the temperature in degrees Kelvin and the coefficients AI through ~ are given byFor t. 5591,2 K
A, = 0.20860000e+04;
CHAP'Tl!R 3
A, ~ O.13530557e+04;
A, ~ -O.33616219e+05;
A., ~ O.53989891e+05;
A, ~ -O.22623269e+05;
A., ~ O.14442905e+04;
A, ~ -O.34480552e+04;
A, ~ 0.47304248e+04;
A, ~ -0. 172409J3e+04;
For t. < 591,2 K
A, ~ O.20860000e+04;
A, ~ O.86537622e+03; A, ~ 0.46032137e+06;
A., ~ -O.7264J812e+06;
A, ~ O.31508275e+06;
A., ~ -0.39173969e+06;
A, ~ O.37372265e+07;
A, ~ -O.25566604e+08;
A, ~ O.77743632e+08;
Enthalpy of sugar cane juice
MODEL DESCRIPTION
The specific enthalpy of sugar cane juice (in kJ/kg) was calculated using the correlation by Lyle (1950)
{(BX) ( 100 +
BX )
hL
=
2,32610
900 _ 8 Bxwhere Bx is the dry solids content (sucrose and impurities) of the sugar cane juice on a mass percent basis and T L is the juice temperature in cc.
CHAP'TER3 MODEL DESCR1PTION
Saturation pressure of water
The absolute vapour pressure of saturated boiling water (in bar) at a given boiling point temperature, Ts in degrees Kelvin, was calculated using the Wagner equation (Wagner,1973):
where
and the coefficients are given by
For 273,2 ,; T,'; 323,2 K
A, = -8,10988775;
A, = 2,17464254;
A, = -3,51897089;
A., = -0,50283681; A, = 647,126;
A =22055"
"<> "
For 323,2 < T, ,; 423,2 K A, = -7,81340301; A, = 1,55097861; A, = -2,84616200;
A., = -1,26127042; A, = 647,126;
A"
= 220,55;+
A
2 '1,1,5 +A
3 '1" +A
4'1'6]
1 - '1'
' 1 ' = 1 - - T,
A s
CHAPTEA3
For 423,2 < T, $ 523,2 K
A, - -7,74037537;
A, - 1,38109682;
A, - -2,60773347;
A. - -1,57952429;
A, - 647,126;
• -22055
"6 "
For T, > 523,2 K
A, - -7,82843137;
A, - 1,69248424;
A, - -4,27203456;
A. -
49,12068163;A, - 647,126;
• - 22055'
"6 "
Saturation temperature of water
MOOEL DESCAIP'TIOH
The boiling point temperature of pure water (in degrees Kelvin) at a given absolute pressure was calculated using the correlation of Aggarwal (1989):
T
=
A + A 2s I logP
.
+ A,
where Ps is the absolute pressure in Mpa and the coefficients are given as:
For P, $ 0,085 MPa
A, - 0.39612064e+02;
A, - -O.39839608e+04;
A, - -0.96562826e+01;
CHAPTER 3
For 0,085 < P, $ 2,625 MPa
AI ~ 0.45864958e+02; A, ~ ·0.38175562e+04;
A, ~ ..Q.93753290e+Ol;
For 2,625 < P, $ 11,25 MPa
AI ~ ..Q.43211877e+01; A, ~ ·0.46978731e+04;
A, ~ ..Q.I0285592e+02;
For p. > 11,25 MPa
AI ~ ..Q.31177350e+03;
A, ~ ..Q.I0757354e+05;
A, ~ ..Q.14310894e+02;
Specific heal capacity of condensale
MOO~L DESCRIFT10N
The specific heat capacity of pure water (in kJ/kg. QC) was calculated using an empirical correlation:
where T c is the temperature of the condensate in
0 c.
Specific heat capacity of sugar cane juice
The specific heat capacity of sugar cane juice (in kJ/kg.°C) was calculated using the correlation developed by Watson (1989):
CPL
=4,1253 - 0,024804Bx + 6,7 x 10-' Bx . TL
+ 1 ,8691
X10-
3TL - 9,271
X10-6
T~CI-lAP'ITR 3 MODEL DESCRIPTION
where T L is the juice temperature and Bx is the dry solids content (sucrose and impurities) of the sugar cane juice on a mass percent basis.
Specific heat capacity of water vapour
The specific heat capacity of water vapour (in kJikg. 'C) was calculated using an empirical correlation:
Cpv
=0,2323418 (8,10 - 0,72 x 10-
3tv + 3,63x 10"'; t~ - 1,16x 10-' t; )
where !v is the vapour temperature in degrees Kelvin.
Swface tension
The swface tension at the liquid / vapour interface in sugar cane juice (in N/m) was calculated using the correlation of Watson (1989):
where T L is the temperature of the juice in °C andBx is the dry solids content (sucrose and impurities) of the sugar cane juice on a mass percent basis.
Thermal conductivity of condensate
The thermal conductivity of pure water (in W/m.'C) was calculated using an empirical correlation:
k
w 6,308 x 10-6 T2 c
where T
c
is the temperature of the condensate inc c.
Thermal conductivity of sugar cane juice
The thermal conductivity of sugar cane juice (in W/m.oC) was calculated using the correlation ofRiedel (1949):
kL =
1,162222 x 10-
3(486 + 1,55T
L -0,005T~)(I- 0,0054Bx)
CI-tAPT!:R 3 MODEL. DescRlPTloN
where T L is the temperature of the juice in °C and Bx is the dry solids content (sucrose and impurities) of the sugar cane juice on a mass percent basis.
Thermal conductivity of water vapour
The thermal conductivity of water vapour (in W Im.oC) was calculated using an empirical correlation:
+ 4 , 3 X 10-4 t2 v -21,73xlO-8 t~) where lv is the vapour temperature in degrees Kelvin.
Viscosity of condensate
The viscosity of pure water (in kg/m.s) was calculated using an empirical correlation:
f'w = 2,73xlO-3 + 2,88xlO--{i Tc - 5,95x10-4logTc
where Tc: is the temperature of the condensate in QC.
Viscos ity of sugar cane juice
The viscosity of sugar cane juice (in cP, where I cP = 1000 kg/m.s) was calculated using the correlation ofGenotelle (1978):