Introduction
Absorption machines have gained increased interest in the recent years. One of the disadvantages of such machine is its large size components especially the absorber which is the most critical component, and its characteristics have significant effect on overall system efficiency. Reduction of the absorber size needs a well investigation of the absorption process inside the absorber. In this aspect, many research have been theoretically conducted (Andberg and Vliet (1987), Jeong and Garimella (2002)) and many experiments has been carried out (Cosenza and Vliet (1990), Hoffmann (1996), Deng and Ma (1999)). Even though an improvement of absorber efficiency, by the development of tubes having higher efficiency and supply of surfactants, was reported (Yoon and Kim (2002), Beutler et al. (1996)), but this machines still facing difficulties to be adopted as a viable residential absorption system. Recently, Garimella (2000) presented and analyzed a miniaturization technology for absorption heat and mass transfer component. He preliminary modeled it for NH3-H2O system and concluded that such concept holds to the potential for the development of extremely small absorption system components. In this paper, the same concept was considered for LiBr-H2O absorbers with different modeling approach, design and simulation.
A full detailed description of the concept can be found in Garimella (2000)
and briefly as follows: Short lengths of small diameter tubes are placed in
square array to form one row, Figure (1.a). The second row
is placed above the first row in a transverse orientation perpendicular to the
tubes in the first row, Figure (1.b). A complete absorber
then can be built in the same manner, Figure (1.c). Cooling
water flows from bottom to the top through all the rows in series form or parallel
through the rows of one pass and then in series through all passes. Strong solution
enters from the top while the vapor enters, countercurrent to the solution,
from the bottom.
|
| Fig. 1: |
Schematic of the concept: (a) one row (b) two rows and (c)
complete absorber |
Modeling and simulation
A mathematical model and computer code were developed to develop and simulate
such absorber or simulate an already existing absorber. In this model, the solution
film flow along one half of the tube is modeled as that along a vertical cooled
wall with a length of half tube circumference, which is a model suggested by
Wassenaar (1995). A schematic representation of the model is shown in Figure
(2). Equations of momentum, energy and diffusion of mass and their specific
boundary conditions for this situation are represented in four dimensionless
combined ordinary differential equations as follows:
|
| Fig. 2: |
Schematic of the model |
These equations describe the average mass fraction of water in the solution
,
the average solution temperature
the heat transferred to the cooling medium across the plate wall per unit width
and
the mass transfer of the water vapor to the film per unit width
in one infinitesimal part of the film with length
as shown in Figure (2). A good estimate for the transfer numbers
defined in Equations (1-4) are, Wassenaar
(1995):
Then, these equations are solved numerically for a unit width of the plate, Khalid and Ali (2001), to give the final form as follows:
where
Absorber design and simulation
The mathematical model developed in the preceding section can be adopted
to design an absorber and simulate an already existing one. In this section
absorber design and simulation are presented, respectively. Utilization of the
developed mathematical model to design an absorber of such type for specific
evaporator cooling capacity, the following design conditions should be given:
| • |
Evaporator load QE |
| • |
System pressure P |
| • |
Solution inlet concentration ξs,i |
| • |
Solution inlet temperature Ts,i |
| • |
Solution outlet concentration ξs,o |
| • |
Solution outlet temperature Ts,o |
| • |
Cooling water inlet temperature Tc,i |
From the evaporator load and system pressure and using steam tables, vapor
flow rate can be calculated. Then, equating this amount of vapor to the vapor
to be absorbed inside the absorber ,
the solution flow rate inlet to the absorber
and solution outlet flow rate can be calculated as follows:
Substituting Equation (11) into Equation
(12) yields:
As first step, general absorber geometry to meat the evaporator load is to be selected:
| • |
Tube outside diameter (do). |
| • |
Tube inside diameter (di). |
| • |
Tube length (L). |
| • |
Number of tubes per row (NT/R). |
| • |
Number of rows per pass (NR/P). |
| • |
Number of passes (NP). |
The algorithm for the solution of equations has constructed in C-Language.
The flow diagram shown in Figure (3) describes the sequence
of various operations. To avoid confusion, the following abbreviations were
used for cooling water routine through rows, passes, and the whole absorber
as shown in Figure (4).
|
| Fig. 3: |
Cooling water routine through (a) row, (b) pass and (c) absorber. |
|
| Fig. 4: |
Schematic of one horizontal tube |
|
: |
Row inlet cooling water temperature, |
|
: |
Row outlet cooling water temperature, |
|
: |
Pass inlet cooling water temperature, |
|
: |
Pass outlet cooling water temperature, |
|
: |
Absorber inlet water temperature, |
|
: |
Absorber outlet water temperature. |
From the above abbreviations it can be seen that
| 1. |
First of all, the necessary parameters and fluid properties
are set. |
| 2. |
As simulation of one tube represents all the tubes in the same row, one
tube is divided along the length into m sections (starting from the
inlet) each of length (dL = L/m). Half circumference of the tube
is divided into n equal parts as shown in Figure (4). |
| 3. |
Starting from the most upper row of the most upper pass and at a particular
section, the inlet conditions are established on the basis of input parameters
such as mass flow rate per unit length per one side of the tube,,
inlet temperature,Τs,i, and concentration,ξs,i,
of the solution. |
| 4. |
The absorber cooling water outlet temperature,,
(which is also the outlet temperature of the most upper pass )
and the pass inlet temperature,,
(very close to the outlet temperature,)
are assumed. |
| 5. |
Under the input conditions to the tube, Equations (6-9)
can be applied to the first part (i = 1) of the first section (j = 1). The
inputs start with
then the outputs are used as inputs to (i = 2) at the same section (j
=1) and so on to the last part (i = n) where the outputs are
All these processes are taking place at constant cooling water temperature
.
|
| 6. |
Solution outlet temperature and concentration for each section is calculated
as follows:
And solution outlet flow rate
|
| 7. |
Steps (5) and (6) are repeated for (j = 2, 3,…, m) with a new cooling
water temperature, ,
at each section. This temperature can be obtained from the heat balance
around the cooling water circuit as follows:
|
| 8. |
Heat transfer to the cooling water from the row, ,
and the total mass flow rate of solution leaving the row, ,
are computed by:
|
| 9. |
Solution temperature and concentration leaving the tube (the row) are
computed by taking the average temperature and concentration over all
sections (j=1,2,…m). |
| 10. |
Steps (5-9) are repeated for the next rows (Ro = 2, 3,… NR/P)
with the above row outlet conditions are used as inputs to the next row
until the last row in the pass is reached. |
| 11. |
Pass cooling water outlet temperature is computed by taking the average
cooling water outlet temperature over all the rows of the pass. At this
stage, if the difference between pass average outlet temperature and the
assumed one is greater than a predetermined value,ξ1, the
cycle is repeated with reducing the previously assumed . |
| 12. |
When the convergence for the upper pass is reached, the whole sequence
of operation proceeds to the next pass with pass cooling water outlet
temperature is set equal to the upper pass inlet temperature. |
| 13. |
Steps (5-12) are repeated until the last pass is reached. At this stage,
absorber cooling water inlet temperature,,
is compared with a specified value, (32°C). If the
difference is greater than a predetermined value,ξ2, the
cycle is repeated with reducing the previously assumed absorber cooling
water outlet temperature,. |
| 14. |
Finally, conditions of the solution leaving the absorber (leaving the
last row), ,
,
and are
compared with the required design ones. If these conditions are different
from the design ones, absorber configuration, (tubes size, tube per unit
row…etc), are changed and the whole sequence is repeated. |
| 15. |
When the resulted outlet conditions are the same as the design ones
the whole results are given and the program terminated. |
This model and computer code was used to develop and simulate an absorber with
the design conditions shown in Table (1.a). The absorber geometry
selected by the code was shown in Table (1.b) whereas the
absorber simulated performance is presented graphically in Figure
(5-8).
| Table 1: |
(a) design conditions and (b) selected absorber geometry |
|
|
Results and discussion:
Figure (5) shows the variation of solution and cooling water
temperatures across the absorber and the corresponding solution concentration
is shown in Figure (6).
|
| Fig. 5: |
Variation of solution and cooling water temperature across
absorber rows |
|
| Fig. 6: |
Variation of solution concentration across absorber rows |
|
| Fig. 7: |
Variation of absorption rate and across absorber rows |
|
| Fig. 8: |
Variation of heat duty across absorber rows |
It can be seen that the solution temperature at the top of the absorber drops significantly as the solution is closed to saturation state, therefore absorption is very small and the cooling water duty is the sensible heat of solution. After this, absorption increases and solution temperature changes tend to stabilize. The cooling water temperature increases from bottom to top and jump periodically every 6 rows due to the change to the next pass. The solution concentration progressively decreases from top to bottom as vapor is absorbed into it.
Variation of vapor absorption rate across the absorber is shown in Figure
(7). At the absorber inlet absorption rate is very small as mentioned before
and then increases as the solution temperature drops and finally decreases uniformly
as the difference between the solution and cooling water temperatures decreases.
Figure (8) shows the variation of heat duty across the absorber.
At the inlet heat duty is high even though absorption rate is small because
of sensible cooling of the solution and then drops toward the bottom with periodic
jump every new pass.
CONCLUSION
A mathematical model and a computer Code for development and simulation of a small-size absorption refrigeration machines using LiBr-H2O as working fluid were developed in this study. Operating conditions were selected as input to this model and code and the corresponding absorber geometry to meet these conditions were given as output. The resulted absorber performance was then simulated and presented graphically.
Nomenclature
|
= |
constants |
|
= |
Biot number |
|
= |
specific heat capacity at constant pressure (kJ/kg °C) |
|
= |
diameter (m) |
|
= |
mass diffusivity of solution (m2/s) |
|
= |
Graetz number |
|
= |
specific enthalpy (kJ/kg) |
|
= |
heat of absorption (kJ/kg) |
|
= |
thermal conductivity (kW/m2 °C) |
|
= |
length (m) |
|
= |
Lewis number |
|
= |
mass flow rate (kg/s) |
|
= |
number |
|
= |
number of tubes |
|
= |
Nusselt number |
|
= |
pressure (kPa) |
|
= |
Prandtl number |
|
= |
heat flow per unit length (kW/m) |
|
= |
Renolds number |
|
= |
Sherwood number |
|
= |
temperature (°C) |
|
= |
mass fraction of water in solution (kg/kg) |
|
|
|
|
= |
average mass fraction |
|
= |
coordinate along wall plate (m) |
|
= |
coordinate perpendicular to the wall plate (m) |
|
|
|
Greek
|
|
= |
dimensionless mass fraction |
|
= |
average dimensionless mass fraction |
|
= |
solution film thickness |
|
= |
dimensionless temperature |
|
= |
average dimensionless temperature |
|
= |
dimensionless heat of absorption |
|
= |
lithium bromide concentration (kg/kg) |
|
= |
dynamic viscosity (kg/m.s) |
|
= |
density (kg/m3) |
|
= |
derivative of with
respect to
at constant ,
∂h/∂w (kJ/kg) |
|
= |
thermal diffusivity (m2/s) |
Subscripts
| = |
sequence number |
|
|
|
|
| = |
absorbed |
|
= |
absorber |
|
| = |
average |
|
= |
cooling water |
|
| = |
equilibrium |
|
= |
interface or inside |
|
| = |
sequence index |
|
= |
number |
|
| = |
entrance or outside |
|
= |
refrigerant |
|
| = |
solution |
|
= |
vapor |
|
| = |
wall |
|
|
|
|
Superscripts