49 CHAPTER 4 MATHEMATICAL MODELING AND SIMULATION 4.1 INTRODUCTION Mathematical modeling is an approach in which, practical processes and systems can generally be simplified through the idealizations and approximations in the form of system of equations to solve a problem. Also, it enables us to understand and predict the behaviour and characteristics of thermal systems. Once a model is formulated, it can be subjected to a range of operating conditions and design variations. This chapter deals with the modeling and simulation of an automobile air-conditioning system operated with R12 and the proposed mixture (M09) as the refrigerant. Automobile air-conditioning systems work basically on the principle of the vapor compression refrigeration cycle. The theoretical vapor compression cycle consists of an isentropic compression, isenthalpic expansion, and isobaric evaporation and condensation. The four major components used are the compressor, thermostatic expansion valve, evaporator and condenser. These components are simulated separately and integrated to simulate the entire system using MATLAB software. REFPROP 7.0 is used to evaluate the refrigerant properties. The simulation model is based on the components actually present in the experimental test rig as detailed in Appendix 1.
Transcript
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CHAPTER 4
MATHEMATICAL MODELING AND SIMULATION
4.1 INTRODUCTION
Mathematical modeling is an approach in which, practical
processes and systems can generally be simplified through the idealizations
and approximations in the form of system of equations to solve a problem.
Also, it enables us to understand and predict the behaviour and characteristics
of thermal systems.
Once a model is formulated, it can be subjected to a range of
operating conditions and design variations. This chapter deals with the
modeling and simulation of an automobile air-conditioning system operated
with R12 and the proposed mixture (M09) as the refrigerant.
Automobile air-conditioning systems work basically on the
principle of the vapor compression refrigeration cycle. The theoretical vapor
compression cycle consists of an isentropic compression, isenthalpic
expansion, and isobaric evaporation and condensation. The four major
components used are the compressor, thermostatic expansion valve,
evaporator and condenser. These components are simulated separately and
integrated to simulate the entire system using MATLAB software. REFPROP
7.0 is used to evaluate the refrigerant properties. The simulation model is
based on the components actually present in the experimental test rig as
detailed in Appendix 1.
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4.2 MODELING AND SIMULATION OF EVAPORATOR
A mathematical model is created to predict the automobile air-
conditioning evaporator performance under steady state conditions. The
evaporator used in the experimental test rig is a fin and tube evaporator,
7 rows deep with three circuits. The total number of tubes is 32 and the
dimensions pertaining to the tube and fin, and the overall dimensions are
detailed in Figure 4.1.
Figure 4. 1 Evaporator tube circuit layout
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4.2.1 Modeling of evaporator
The following assumptions have been made in simulation of evaporator:
Refrigerant flow was one-dimensional.
The pressure drop measured across the evaporator is uniformly distributed.
The heat transfer from refrigerant to atmosphere through the bend regions was negligible.
The total air delivered by the fan was equally divided over the entire tube length.
Pressure of vapor and liquid was equal at all points in the cross section of any tube.
The effect of the oil present in the refrigerant is negligible.
The tubes are either fully dry or fully wet.
The water film on the wet nodes is assumed to be of negligible
thickness, and the heat carried away by the drainage of the
condensate is ignored.
The refrigerant side and airside are modeled separately. Further, the
evaporator is divided into a superheated region and two-phase region. When
the evaporator performs sensible cooling, the surface temperature is used to
estimate the heat transfer between the refrigerant and the air. When the
evaporator is cooling and dehumidifying, the water film (condensate)
temperature on the coil surface is used for calculating the heat transfer.
Though the evaporator tube layout is a cross counter flow arrangement, and
the number of rows is seven, it can be assumed to be a counter flow
arrangement (more than three rows can be assumed as counter flow –
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(Mcquistion and Parker 1994, Stevens 1957, Kays and Crawford 1993). The
entire tube length of the evaporator is segregated into small control volumes
of known length (2 mm) as shown in Figure 4.2 (Vardhan 1998).
Figure 4.2 Single tube - counter flow heat exchanger
Figure 4.3 shows the dry and wet zones for airside, and saturated
and superheated region for the refrigerant side. When the analysis is done in
the direction of airflow, control volume ‘1’ will be the first segment of contact
for air and it will be the last segment of contact for refrigerant. For the
refrigerant, the inlet pressure is given and with the pressure drop already
known from the experimental study, the outlet pressure can be calculated.
Since, the evaporator superheat is initialized first, the refrigerant outlet
temperature is also known. The evaporator inlet air temperature is the return
air temperature from the cabin, which is held constant at 27C. Therefore, Tai
and Tro as seen in Figure 4.3 are known. From these temperatures and the
available geometry of the heat exchanger, the surface temperature can be
found by establishing a heat balance between air and refrigerant side. From
the surface temperature, the heat transferred in that control volume and the
outlet conditions can be determined. These outlet conditions will be supplied
as the inlet conditions for the second control volume. The surface temperature
is compared with the dew point temperature of the air at the evaporator inlet.
If the surface temperature is above the dew point temperature, condensation
of moisture will not take place. This procedure is repeated for other control
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volumes until the surface temperature is less than dew point temperature of
the entering air. The length required for the superheated region is calculated.
Figure 4.3 Dry and wet zones in the counter flow heat exchanger
From this portion of the evaporator instead of temperature
difference as the driving potential, the enthalpy difference is used as the
driving potential. The maximum enthalpy difference is the difference between
the enthalpy of air at the point of condensation and the enthalpy of the
saturated air corresponding to the refrigerant inlet conditions. Now, for the
remaining wet region the ε-NTU method is used for the calculation of the
evaporator duty, but with the modified heat transfer coefficient, which
includes the condensation of moisture in the air (Equation 4.27).
For a counter flow heat exchanger the effectiveness can be related
to the number of transfer units (NTU) with the following expression (Kays
and London 1964)
(1 *)
(1 *)
11 *
NTU C
NTU C
eC e
(4.1)
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min
max
* CCC
(4.2)
In the two phase region, the heat capacity on the refrigerant side
approaches infinity and the heat capacity ratio C* tends to zero, the
effectiveness for any heat exchanger in the two phase region is expressed as
(Kays and London 1964)
1 NTUe (4.3)
The equations 4.11 to 4.27 used to calculate the heat exchanger
parameters below are referred from Mcquistion and Parker (1994) and
Kuppan (2003). The NTU is a function of the overall heat transfer coefficient
and is defined as
min
a aU ANTUC
(4.4)
The overall heat transfer coefficient accounts for the total thermal
resistance between the two fluids. Neglecting the fouling resistance, it is
expressed as follows (Mcquistion and Parker 1994)
1
1 1a a w
sa a a r r
U A RA A
(4.5)
ln
2
oa
iw
DAD
RkL
(4.6)
a p finA A A (4.7)
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The surface efficiency on the refrigerant side is considered to be
unity as there are no fins. To calculate the fin efficiency on the airside, it is
necessary to find the equivalent radius of the fin. The empirical relation for
the equivalent diameter is given by McQuiston and Parker (1994).
121.27 ( 0.3)eq
i
DD
(4.8)
The coefficients Ψ and β are defined as
Mr
(4.9)
LM
(4.10)
Once the equivalent radius had been determined, the equations for
the standard circular fins were used. The length of the fin was much greater
than the fin thickness. Therefore, the standard extended surface parameter, mes
can be expressed as,
2 aes
fin fin
mk t
(4.11)
For circular tubes a parameter Φ can be defined as
1 1 0.35lnR Rr r
(4.12)
The fin efficiency, ηfin for a circular fin is a function of mes, Deq and
Φ and can be expressed as
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tanh( )esfin
es
m rm r
(4.13)
The total efficiency of the fin ηsur, is therefore expressed as
1 1finsur fin
a
AA
(4.14)
After finding the overall heat transfer coefficient, NTU is
determined and from that the efficiency was evaluated. In general, the heat
transfer rate is computed using,
Q m h (4.15)
The effectiveness is expressed by
max
QQ
(4.16)
The maximum heat that could be transferred is given by, the
product of the minimum heat capacity and inlet temperature difference of the
two fluids.
max min ri aiQ C T T (4.17)
The actual heat transferred is given by
min ri aiQ C T T (4.18)
The surface temperature is calculated by equating the airside heat
transfer rate and the refrigerant side heat transfer rate and can be expressed as,
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ha o a s r r s rh A T T h A T T (4.19)
ha a a a r r rs
ha a a r r
h A T h A TTh A h A
(4.20)
The heat transferred in the superheated region when the surface
temperature is more than the dew point temperature i.e., under non-
dehumidifying conditions, is expressed as (Wang 1990).
minsh aei dpQ C T T (4.21)
The heat transferred in the superheated region when the surface
temperature is less than the dew point temperature, i.e., under dehumidifying
conditions, is expressed as,
( )sh aei sriQ m h h (4.22)
The standard extended surface parameter, mes for a fin under
dehumidifying conditions can be expressed as, (McQuiston and Parker 1994).
2 1 fg a soes
fin fin a a s
h W Wm
k t Cp T T
(4.23)
The outlet enthalpy of the dehumidifying air can be calculated from
(Wang 1990)
( )ao ai aei srih h h h (4.24)
4.2.2 Evaporator - Heat transfer correlation
The mathematical model needs to be supplemented with heat
transfer coefficient correlations for both the fluids involved in the heat
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transfer. The airside heat transfer correlations and the refrigerant side heat
transfer correlations are presented in this section.
4.2.2.1 Air side heat transfer correlations
The air side convective heat transfer coefficient is expressed as
23Cph = j * G *Cp *
Ka a
sa a aa
(4.25)
where, 0.502 0.0312
0.3280.14Re ta
l o
TP FSjTP D
(4.26)
The heat transfer coefficient of the humid air is given by (Liang 1999)
( )
1( )
fg sha sa
a a s
h W Wh h
Cp T T
(4.27)
when there is no dehumidification, then (W-Ws) will be zero and hha = has.
4.2.2.2 Refrigerant side heat transfer correlations
The heat transfer coefficient in the single-phase region is given by
the Dittus-Boelter equation.
0.8 0.4*0.023Re Prvsh
hr
KhD
(4.28)
The heat transfer coefficient in the two-phase region is given by the
Klimenko equation (Castro et al 1993).
0.2 0.09
0.6 1.60.087 Re Pr v a ltp eq l
l l
K KhK LC
(4.29)
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where, Pr l ll
l
CpK
(4.30)
and equivalent Reynolds number is given by
Re = 1 1lreq i
ff v l
m LCxA
(4.31)
where, LC is the Laplace constant.
( )
l
l v
LCg
(4.32)
4.2.3 Simulation of the evaporator
The complete simulation procedure of the evaporator is summarized below. Also, a flow chart depicting the algorithm is detailed in Figure 4.4.
Input evaporator dimensions, mass flow rate, Pci, Pei
Initialize Qsum, SH, SC, segmental length, etc.
Calculate Surface geometrical parameters.
Use section-by-section scheme.
Calculate air and refrigerant thermo-physical properties.
Calculate dimensionless parameters Reynolds number, Prandtl number etc.
Calculate j for air and Nu for refrigerant.
Check whether surface temperature is less than dew point temperature.
Calculate: μ, ρ, Cp, k, Re, Pr, Single phase sensible ht tr. coeff. –
Dittus Boelter equation Cmax, Cmin, ηf , U ε, NTU, Q Pressure drop for the segment (Δpseg) Pi = Pi + Δpseg, Lseg = Lseg + dl
While Tsur > = Tdp
Qsum = Qsum + Qsh
Calculate: μ, ρ, Cp, k, Re, Pr, Single phase sensible ht tr coeff. –
Dittus Boelter equation Cmax, Cmin, ηf , U ε = ma (enthalpy diff), NTU, Q Pressure drop for the segment (Δpseg) Pi = Pi + Δpseg, Lseg = Lseg +dl
While Tsur > = Tdp & Tref <Tsat
Qsum = Qsum + Qsh Super heated
Region
61
Figure 4.4 Flow chart for Evaporator simulation (continued)
Two Phase Region
Calculate: μ, ρ, Cp, k, Re, Pr, j – factor, Heat transfer coefficient – humid air
Calculate: Heat absorbed for 0.01 quality rise
Qtp = f (Pi, xi, xo) = mf * ΔH μ, ρ, Cp, k, σ, Re, Pr, for saturated
liquid and vapour and Laplace constant
Two phase ht.tr. coeff.– Klimenko eqn Cmax, Cmin, ηf , U ε = ma (enthalpy diff), NTU, Q Pressure drop for the segment (Δpseg) Pi = Pi – Δpseg, Lseg = Lseg +dl
While X < = 1
Qsum = Qsum + Qtp
xi = xo xo = xo + 0.01
Check Tassumed = Tout
Print outputs:
Stop
A
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If ‘the condition is not satisfied’ – find sensible heat transfer
coefficient. If the condition is satisfied –find combined heat
transfer coefficient and proceed.
Determine fin efficiency.
Calculate the overall conductance (1/UA).
Using ε - NTU method, find effectiveness.
Calculate the heat transferred in one section.
Calculate the pressure drop in that section.
Check for surface - dew point temp and saturation
temperature.
Once dew point is reached, assume exit temperature of air.
Calculate enthalpy difference instead of temperature
difference to analyse the heat transfer in the wet region.
Proceed to the next section till saturation temperature is
reached
Find the cumulative length required for the superheat region.
Vary quality by 0.1.
Calculate the heat transferred and the pressure drop for the
remaining length.
Calculate the outlet quality and total heat transferred.
Find the outlet temperature of the air and check with the
assumed temperature. Substitute successively till the
convergence is achieved.
The deliverables from the evaporator outlet are evaporator
outlet temperature, quality and evaporator heat transfer rate.
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4.3 MODELING AND SIMULATION OF CONDENSER
4.3.1 Modeling of condenser
The following assumptions have been made in simulation of condenser:
The pressure drop in the straight tubes is uniform throughout.
Refrigerant flow was one-dimensional.
The heat transfer from refrigerant to atmosphere through the
bends is negligible.
The total air delivered by the fan was equally divided over the
entire tube length.
Pressure of vapor and liquid was equal at all points in the
cross section of any tube.
The effect of the oil present in the refrigerant is negligible.
A mathematical model is created to predict the automobile air-
conditioning condenser performance under steady state conditions. The
condenser used in the experimental test rig is a flat tube serpentine condenser
with louver fins and cross flow arrangement. The total number of tubes is 14
and the dimensions pertaining to the tube and fin and the overall dimensions
are detailed in Figure 4.5. The refrigerant side and airside are modeled
separately. Further, the condenser is divided into a de-superheating region, a
two-phase region and a sub-cooled region. The entire tube length of the
condenser is separated into small control volumes of known length (2 mm).
4.3.2 Condenser - Heat transfer correlation
The airside heat transfer coefficient is given by (McQuiston and
Parker, 1994).
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Figure 4.5 Cut view of the flat tube automotive condenser with louver
fins
23Cph = j * G *Cp *
Ka a
a a aa
(4.33)
Re a aa
a
G Dh
(4.34)
The colbourn factor is given by (Castro and Ali, 2000)
2Re Rej = 0.02633 - 0.02374 * + 0.007383 *
2000 2000a a
(4.35)
The heat transfer coefficient of the superheated region is given by
(Dittus and Bolter equation)
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0.8 0.3*0.023Re Prvsh
hr
KhD
(4.36)
The heat transfer coefficient in the two-phase region is given by
(Castro and Ali, 2000)
0.8 0.3tp
Kh = * 0.05 Re Pr D
leq l
hr
(4.37)
0.5
Re Re Rev leq v l
l v
(4.38)
The overall heat transfer coefficient is calculated from
1
1 1a a w
sa a a r r
U A RA A
(4.39)
ln
2
oa
iw
DAD
RkL
(4.40)
The overall surface area is given by
a p finA A A (4.41)
The other heat exchanger relations discussed in the evaporator
holds good for the condenser too (except dehumidifying correlations).
4.3.3 Simulation of the condenser
The complete simulation procedure of the condenser is summarized
below. A flow chart depicting the calculation procedure is detailed in
Figure 4.6.
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Input condenser dimensions; mass flow rate, Pci, Pei.
Calculate Surface geometrical parameters.
Use section-by-section scheme.
Calculate air and refrigerant thermo-physical properties.
Calculate: μ, ρ, Cp, k, Re, Pr, j – factor, Heat transfer coefficient – dry air
A
Calculate: Heat absorbed for 0.01 temperature drop
Qsh = f (Pi, ti, to) = mf * ΔH μ, ρ, Cp, k, Re, Pr, Single phase ht tr coeff. – Dittus Boelter Cmax, Cmin, ηf , U ε, NTU, Q Pressure drop for the segment (Δpseg) Pi = Pi – Δpseg, Lseg = Lseg + dl
While Ti < = Tsat
Increment segmental length - dl
Qsum = Qsum + Qsh
Ti = To To = To - 0.01
Check Q>Qsh
Super heated Region
69
Figure 4.6 Flow chart for Condenser simulation (Continued)
Calculate: Heat absorbed for 0.01 quality drop
Qtp = f (Pi, xi, xo) = mf * ΔH μ, ρ, Cp, k, σ, Re, Pr, for saturated
liquid and vapour Two phase ht. tr. coeff. - Ali eqn. Cmax, Cmin, ηf , U ε, NTU, Q Pressure drop for the segment (Δpseg) Pi = Pi – Δpseg, Lseg = Lseg + dl
While X > = 1
Qsum = Qsum + Qtp
xi = xo xo = xo - 0.01
Check Q>Qtp
A
Calculate: Heat absorbed for 0.01 temperature
drop, Qsh = f (Pi, ti, to) = mf * ΔH μ, ρ, Cp, k, Re, Pr, Single phase ht. tr. coeff. – Dittus
Boelter Cmax, Cmin, ηf , U ε, NTU, Q Pressure drop for the segment (Δpseg) Pi = Pi – Δpseg, Lseg = Lseg +dl
While Lcum < =TL 1
Qsum = Qsum + Qsh
Ti = To To = To – 0.01
Check Q>Qsc
Print outputs: Ts,Taeo,Qe,
Stop
Two Phase Region
Sub cooled Region
A
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4.4 MODELING AND SIMULATION OF COMPRESSOR
4.4.1 Modeling of compressor
The following assumptions are made in the simulation of
compressors.
The modeled compressor cycle is an approximation of a real
compressor cycle.
Compression and expansion are assumed to be polytropic.
The polytropic exponent is a function of the refrigerant type
and compression ratio.
The lubricant oil has negligible effects on the refrigerant
properties.
The pressure loss in the valves, and pipelines, are negligible.
A compressor was modeled as a volume flow device, by using the
basic thermodynamic equations. The low-pressure superheated refrigerant
vapor coming out of the evaporator is compressed to the condenser pressure
in the compressor. A set of equations and empirical relations were used to
model the compression process. The compressor used in the present work is a
swash plate type with a displacement volume of 108 cc per revolution of the
compressor shaft. A swash plate compressor usually has 5 to 9 cylinders with
individual pistons on one side or on either side of the connecting rod.
The model was assumed to be a single cylinder compressor whose
swept volume equals the total swept volume of the five-cylinder compressor.
The inputs to the compressor are the evaporator outlet pressure, temperature
and compressor speed. The volumetric efficiency of the compressor will be
high at low speeds and low at high speeds. Since an automobile air-
conditioning system is tested for different speeds, an empirical equation for
volumetric efficiency, as a function of speed and pressure ratio, is obtained by
the curve fitting the experimental data. A flowchart detailing the calculation
