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Superheating of ice slurry in melting heat exchangers
international journal of refrigeration 31 (2008) 911–920
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Superheating of ice slurry in melting heat exchangers P. Pronk*, C.A. Infante Ferreira, G.J. Witkamp Process and Energy Department, Delft University of Technology, Leeghwaterstraat 44, 2628 CA, Delft, The Netherlands
article info
abstract
Article history:
One of the main components of an ice slurry system is the melting heat exchanger, in
Received 24 September 2006
which ice slurry absorbs heat resulting in the melting of ice crystals. Design calculations
Received in revised form
of melting heat exchangers are mainly based on heat transfer and pressure drop data,
4 September 2007
but recent experimental studies have shown that superheating of ice slurry should also
Accepted 16 September 2007
be considered. This paper presents ice slurry melting experiments with a tube-in-tube
Published online 22 November 2007
heat transfer coil. The experimental results indicate that operating conditions such as ice slurry velocity, heat flux, solute concentration, ice fraction, and ice crystal size deter-
Keywords:
mine the degree of superheating. The various influences are explained by considering
Ice slurry
the melting process as a two-stage process consisting of the heat transfer between wall
Heat exchanger
and liquid and the combined heat and mass transfer between liquid and crystals. Bigger
Melting
ice crystals and higher solute concentrations decrease the rate of the second stage and
Experiment
therefore increase the degree of superheating. ª 2007 Elsevier Ltd and IIR. All rights reserved.
Parameter Heat transfer Mass transfer Superheating
Surchauffe du coulis de glace dans les e´changeurs de chaleur avec fonte de glace Mots cle´s : Coulis de glace ; E´changeur de chaleur ; Fusion ; Expe´rimentation ; Parame`tre ; Transfert de chaleur ; Transfert de masse ; Surchauffe
1.
Introduction
In the last 10 years, ice slurry has successfully been applied as secondary refrigerant in refrigeration and air-conditioning installations. The application of ice slurry minimizes the required amount of primary refrigerant and enables thermal storage. The latter makes it possible to produce ice slurry at
night with the benefits of low electricity tariffs and low condensing pressures, and use it in daytime when cooling loads normally peak. Proper application of thermal storage leads to a reduction of investment and/or energy costs. After production and storage, ice slurry is transported to applications where it melts and provides cooling for rooms, products or processes. In general, two different methods of ice
* Corresponding author. Tel.: þ31 15 2784894; fax: þ31 15 2786975. E-mail address: [email protected] (P. Pronk). 0140-7007/$ – see front matter ª 2007 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2007.09.008
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Nomenclature A BA BV c1.3 cp dFeret d D G h Dhf k m _ m N n Nucr Dp Pr Q_ Sc Shcr T DTln DTsh t Dt U u V_ w
area (m2) area shape factor volume shape factor constants specific heat (J/kg K) average crystal Feret diameter (m) tube diameter (m) diffusion coefficient (m2/s) growth rate (m/s) enthalpy (J/kg) latent heat of fusion of ice (J/kg) mass transfer coefficient (m/s) mass (kg) mass flow (kg/s) number of crystals number of measurements crystal Nusselt number, acrdFeret/lliq pressure drop (Pa) Prandtl number, cp,liqmliq/lliq heat (W) Schmidt number, mliq/(rliqD) crystal Sherwood number, kdFeret/D temperature ( C) logarithmic mean temperature difference (K) degree of superheating (K), defined in Eq. (11) time (s) measurement interval (s) overall heat transfer coefficient (W/m2 K) velocity (m/s) volume flow rate (m3/s) solute mass fraction
slurry melting are used. In the first method, ice slurry is directly poured onto food products for example fish, resulting in high cooling rates and therefore high product qualities (Fikiin et al., 2005; Torres-de Marı´a et al., 2005). In the second method, ice slurry absorbs heat in regular heat exchangers cooling a second fluid, for example air in display cabinets for supermarkets or air-conditioning systems. Design calculations for this kind of heat exchangers are mainly based on experimentally obtained heat transfer and pressure drop data for melting ice slurry (Ayel et al., 2003; Egolf et al., 2005; Lee et al., 2006; Niez_ _ goda-Zelasko, 2006; Niezgoda-Zelasko and Zalewski, 2006; Stamatiou and Kawaji, 2005). Recent experimental studies have shown that ice slurry may be significantly superheated at the outlet of melting heat exchangers (Hansen et al., 2003; Kitanovski et al., 2005; Frei and Boyman, 2003). This phenomenon should also be taken into account in design calculations, since superheating increases the average ice slurry temperature in the heat exchanger to a value that is higher than expected from equilibrium calculations. As a result, the temperature difference between ice slurry and the other fluid is lower resulting in reduced heat exchanger capacities. Ice slurry is considered superheated when its liquid temperature is higher than its equilibrium temperature.
w0 Dx
initial solute mass fraction in liquid length of control volume
Greek a z l m r s f
heat transfer coefficient (W/m2 K) relative superheating, defined in Eq. (12) thermal conductivity (W/m K) viscosity (Pa s) density (kg/m3) period (h) ice mass fraction
Subscripts cr crystal EG ethylene glycol solution eq equilibrium eqin equilibrium at inlet fr freeze point he heat exchanger i inside ice ice in inlet init initial inner inner is ice slurry liq liquid meas measured o outside out outlet real real rest other components stor storage w wall
Superheating can be explained by considering the melting process of ice slurry as a two-stage process. First the heat exchanger wall heats the liquid and subsequently the superheated liquid melts the ice crystals. The relation between the rates of both processes determines the degree of superheating. For example, when crystal-to-liquid heat and mass transfer processes are relatively slow compared to the wallto-liquid heat transfer process, the degree of superheating is high. The degree of superheating is expected to depend on operating conditions, such as ice slurry velocity, heat flux, solute concentration, ice fraction and ice crystal size. However, quantitative data on superheating are only reported by Frei and Boyman (2003) indicating that the degree of superheating increases as the ice fraction decreases. Moreover, the physical mechanisms that determine the degree of superheating are not understood yet. The aims of this paper are therefore to investigate the influence of operating conditions on the degree of superheating and to unravel the physical mechanisms behind this phenomenon. For this purpose, experiments on ice slurry melting have been performed in a tube-in-tube heat transfer coil, during which the influences of various operating conditions on superheating have been investigated.
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2.
Experimental method
Table 1 – Experimental conditions of melting experiments
2.1.
Experimental set-up
Exp. no.
The melting experiments were performed with the experimental set-up shown in Fig. 1. A fluidized bed heat exchanger as described by Pronk et al. (2006) was used to produce ice slurry from aqueous sodium chloride solutions. The produced ice slurry was stored in an insulated tank that was equipped with a mixing device to keep the ice slurry homogeneous. The tank could easily be disconnected from the set-up and be placed in a cold room for isothermal storage. After production and eventually isothermal storage, the ice crystals were analyzed with a visualization section consisting of a flow cell and a microscope. The ice slurry was subsequently pumped through the inner tube of a tube-in-tube heat transfer coil, which had an internal diameter of 7.0 mm, an outside diameter of 9.5 mm and a total external heat-exchanging surface of 0.181 m2. A 20 wt% ethylene glycol solution extracted from a thermostatic bath flowed counter currently through the annulus with a hydraulic diameter of 6.2 mm and heated the ice slurry in the inner tube. The melting process was continued until all ice crystals were molten and the tank contained only liquid. PT-100 elements with an accuracy of 0.01 K measured the temperatures at the inlets and outlets of the heat exchanger. The mass flow of ice slurry was measured using a coriolis mass flow meter and a magnetic flow meter was used to measure the flow rate of the ethylene glycol solution. The coriolis mass flow meter was also able to measure the temperature of ice slurry downstream of the heat exchanger. All flow rates and temperatures were automatically measured every 10 s, with the exception of the temperature measured in the coriolis mass flow meter which was manually recorded.
2.2.
Experimental conditions
The experimental series consisted of 10 melting experiments, in which the operating conditions were systematically varied as shown in Table 1. The varied conditions were the ice slurry velocity, the heat flux, the ice crystal size and the sodium chloride concentration. The heat flux was adjusted by varying the
1 2 3 4 5 6 7 8 9 10
Tfr ( C)
uis (m/s)
TEG,in ( C)
sstor (h)
6.6 6.6 6.6 6.6 6.6 3.5 11.0 7.1 7.0 7.1
4.1 4.1 4.1 4.1 4.1 2.1 7.4 4.4 4.4 4.4
1.0 1.5 2.0 2.5 1.5 1.5 1.5 1.8 1.7 1.8
3.0 3.0 3.0 3.0 3.0 5.2 0.7 2.6 2.6 0.0
0 0 0 0 16 0 0 0 15 0
dFeret (mm) 249a 249 249a 249a 283 338 133 148 277 148b
fin;init (wt%) 17 18 18 16 16 14 17 10 9 10
a Assumed equal as in experiment 2. b Assumed equal as in experiment 8.
inlet temperature of the aqueous ethylene glycol solution. In most experiments, the difference between the initial freezing temperature of the aqueous solution and the inlet temperature of the ethylene glycol solution was 7.1 0.2 K, except for experiment 10 where this temperature difference was only 4.4 K. In this last experiment the heat flux varied from 4 to 7 kW/m2, while the heat flux in the other experiments was from 7 to 13 kW/m2. The average crystal size was determined by analyzing the crystals with a visualization section. In this respect, the Feret diameter was used as characteristic crystal size, which is defined as the diameter of a circle with the same area as the projection of the crystal (Pronk et al., 2005a). Ice crystals produced from aqueous solutions with equal solute concentration and equal production procedure appeared to have approximately the same average crystal size. The average crystal sizes at the start of experiments 1, 3 and 4 were therefore assumed equal to the average crystal size determined at the start of experiment 2. The same assumption was made for the crystal sizes of experiments 8 and 10. Ice crystals produced from aqueous solutions with higher solute concentrations appeared to have smaller crystals. In order to vary the average crystal size for a certain solute concentration, ice slurry was isothermally stored in the cold room. During isothermal storage, the average crystal size increased as a result of Ostwald ripening (Pronk et al., 2005a,b). In general, the 10 melting experiments listed in Table 1 all showed the same trends on superheating. Therefore the observed phenomena are discussed for one experiment only, namely experiment 1. Subsequently, the results of the different experiments are compared.
3.
Fig. 1 – Schematic overview of the experimental set-up.
w0 (wt%)
Results: analysis of a single experiment
At the start of experiment 1, the ice fraction at the inlet was 17 wt% while the temperature was 5.0 C (see Fig. 2). According to the heat balance, the reduction in ice fraction was initially approximately 9 wt% per pass, which resulted in an expected outlet ice fraction of about 8 wt% and an expected outlet temperature below the initial freezing temperature of 4.1 C. However, the measured outlet temperature exceeded this initial freezing temperature by about 1 K, which means
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Fig. 2 – Measured ice slurry temperatures and ice fraction at the inlet based on equilibrium during melting experiment 1.
that the ice slurry at the outlet was superheated. The temperature measured in the coriolis mass flow meter was below the measured outlet temperature, which is attributed to the release of superheating downstream of the heat exchanger.
3.1.
Assuming equilibrium at the inlet
If it assumed that ice slurry entering the heat exchanger is in equilibrium, then the ice fraction at the inlet is calculated by fin;eqin ¼
win;eqin w0 win;eqin
(1)
In Eq. (1), the weight fraction of solute in the solution win,eqin is determined from the measured inlet temperature assuming equilibrium: (2) win;eqin ¼ weq Tin;meas The development of the inlet ice fraction fin,eqin shown in Fig. 2 suggests that all ice crystals had melted at t ¼ 2400 s. However, by that time ice crystals were still observed in the ice suspension tank. Another indication that ice crystals were still present in the system is that the rate of change of the measured inlet temperature does not change significantly at t ¼ 2400 s. However, a considerable difference is observed at t ¼ 3200 s indicating that all ice crystals had melted by that time. The described observations indicate that ice slurry is also not in equilibrium at the inlet of the heat exchanger, at least during the final stage of the experiment. For a correct analysis, ice slurry should therefore be considered as a non-equilibrium fluid at both inlet and outlet.
3.2.
his;in;eqin ¼ 1 fin;eqin hliq win;eqin ; Tin;meas for Tin;meas Tfr þ fin;eqin hice Tin;meas
ð3Þ
his;in;eqin ¼ hliq w0 ; Tin;meas for Tin;meas > Tfr
(4)
For temperatures above the freezing temperature, the enthalpy simply equals the enthalpy of the aqueous solution. At temperatures below the freezing temperature, the enthalpy of ice slurry is the weighted average of the enthalpy of the solution and the enthalpy of ice. The enthalpy based on equilibrium calculations can be compared with the enthalpy based on the cumulative heat input, which consists of the heat transferred in the heat exchanger and the heat input by other components, such as the pump and the mixing device in the tank: Z t _ Q he þ Q_ rest Þ vt (5) his;in;real ftg ¼ his;in;eqin ft ¼ 0g þ mis 0 The enthalpy at t ¼ 0 is determined by assuming that ice slurry is in equilibrium at the beginning of the experiment. The integral in Eq. (5) is rewritten into a summation in order to apply it to the measured data: his;in;real ftg ¼ his;in;eqin ft ¼ 0g þ
n¼t=Dt X n¼0
Q_ he þ Q_ rest ÞDt mis
The heat input by the other components is determined by equating the enthalpy based on equilibrium with the enthalpy based on the temperature measurement for the final measurement at t ¼ 3400 s (see Fig. 3). By this time, no ice crystals were present in the tank and therefore the enthalpy based on the temperature represents the real enthalpy. Application of this method to melting experiment 1 results in a heat input by the other components of 170 W. Heat input values calculated for the other experiments showed similar numbers. Since the real enthalpy at the inlet of the heat exchanger is known from Eq. (6), the enthalpy at the outlet can be calculated by his;out;real ¼ his;in;real þ
Q_ is _ is m
Assuming non-equilibrium at the inlet
In order to quantify superheating at both the inlet and outlet of the heat exchanger, the enthalpy of ice slurry at both locations is considered. First, the enthalpy at the inlet is considered for the assumption of equilibrium at this location:
(6)
Fig. 3 – Ice slurry enthalpies at the inlet during melting experiment 1.
(7)
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When ice slurry is not in equilibrium, the ice fraction cannot be calculated by using the initial solute concentration w0 and the measured temperature only (see Eqs. (3) and (4)). The nonequilibrium state requires a third thermodynamic property to calculate the ice fraction, for example enthalpy. The enthalpy of ice slurry that is not in equilibrium is given by: (8) his;real ¼ ð1 freal Þhliq fwreal ; Tmeas g þ freal hice Teq fwreal g At the inlet and outlet of the heat exchanger, the enthalpies are known from Eqs. (6) and (7) and the temperature is known from measurements. Thus Eq. (8) contains only two unknown variables, namely the solute concentration in the solution wreal and the ice fraction freal. Since the ice slurry is homogeneously mixed in the tank, it is assumed that the solute concentration in the slurry always equals the initial solute concentration w0. The ice fraction freal is therefore directly related to the solute concentration in the solution wreal by means of the solute mass balance: w0 ¼ ð1 freal Þwreal
(9)
The ice fractions at the inlet and outlet can now be solved iteratively from Eqs. (8) and (9), which is shown for experiment 1 in Fig. 4. The solute concentration in the liquid wreal is subsequently used to calculate the equilibrium temperature at the inlet and outlet of the heat exchanger: Teq ¼ Teq fwreal g
(10)
Fig. 5 shows that the measured outlet temperatures exceed the calculated equilibrium temperatures at the outlet indicating that the ice slurry is considerably superheated at this location.
3.3.
Superheating definition
In order to quantify superheating at the outlet of the heat exchanger, the degree of superheating DTsh is defined as the difference between the measured and the equilibrium temperature: DTsh ¼ Tliq;meas Teq fwreal g
(11)
The melting of ice slurry in a heat exchanger can be considered as a process consisting of two stages as shown in Fig. 6.
Fig. 5 – Measured ice slurry temperature, and calculated ice fractions and equilibrium temperatures at the outlet of the heat exchanger during melting experiment 1.
The first stage consists of the heat transfer process from the wall to the liquid. The driving force for this process is the temperature difference between the wall and the liquid phase. The second stage is the actual melting of the ice crystals, where the difference between the liquid temperature and the equilibrium temperature, hence the degree of superheating, is the driving force. The degree of superheating can therefore be seen as a fraction of the total driving force of the melting process: z¼
Tmeas Teq DTsh ¼ DTw-liq þ DTsh Tw Teq
(12)
This relative superheating z enables the comparison of the superheating results from experiments with different mass flow rates and different heat fluxes. For the analysis of the relative superheating, it is necessary to calculate the wall temperature at the ice slurry side of the inner tube. Here, the ratio of heat transfer coefficients, the ice slurry temperature and the temperature of the ethylene glycol solution are used to determine this temperature: ðTw Tis Þ Uo do;inner ¼ ðTEG Tis Þ ai di;inner
(13)
The overall heat transfer coefficient Uo is deduced from the total heat flux in the heat exchanger, which was determined
Fig. 4 – Measured ice slurry temperatures and calculated ice fractions during melting experiment 1.
Fig. 6 – Schematic representation of temperatures during melting of ice slurry in a heat exchanger.
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from the flow rate and the inlet and outlet temperatures of the ethylene glycol solution: Q_ he ¼ Uo Ao DTln
(14)
The logarithmic temperature difference in Eq. (14) is calculated from the measured temperatures at the inlets and outlets of the heat exchanger. The overall heat transfer resistance (1/Uo) consists of three parts, the annular side heat transfer resistance, the heat transfer resistance of the tube wall, and the wall-to-liquid heat transfer resistance: 1 1 do;inner do;inner 1 do;inner ¼ þ ln (15) þ Uo ao ai di;inner 2lw di;inner The Wilson plot calibration technique was used to formulate single-phase heat transfer correlations for both sides of the heat exchanger. The correlation for the annular side was used to calculate the heat transfer coefficient of the ethylene glycol flow ao. Subsequently, this heat transfer coefficient is used to determine the heat transfer coefficient for the ice slurry flow ai from Eq. (15). Finally, both the overall heat transfer coefficient and the ice slurry heat transfer coefficient are used to calculate the wall temperature from Eq. (13).
4.
Results: influence of operating conditions
4.1.
Influence of ice fraction and ice slurry velocity
the relative superheating of the experiments with slurry velocities of 1.0, 1.5 and 2.0 m/s are very similar, but that the relative superheating at a velocity of 2.5 m/s is slightly lower.
4.2.
The results from the experiments with different ethylene glycol solution inlet temperatures in Fig. 8a show that the degree of superheating increases as the heat flux increases. However, the relative superheating z is similar for different heat fluxes as shown in Fig. 8b.
4.3.
Influence of crystal size
The results of experiments 8 and 10 in Fig. 8a indicate that ice slurries consisting of larger crystals exhibit higher degrees of superheating. Accordingly, the relative superheating also increases as the average ice crystal size increases (see Fig. 8b). A comparison of the superheating results of experiments 2 and 5, in which the crystal size was also the only varied variable, leads to the same conclusion.
4.4.
The superheating results for different ice slurry velocities in Fig. 7a clearly show that the degree of superheating increases as the ice fraction decreases. The figure also shows that for ice fractions higher than 5 wt%, the degree of superheating is higher in the experiments with low ice slurry velocities. This higher degree of superheating is mainly the result of the higher wall temperature caused by the lower wall-toliquid heat transfer coefficient at low slurry velocities. The results for the relative superheating z in Fig. 7b take these different wall temperatures into account. This figure shows that
Influence of heat flux
Influence of solute concentration
The superheating results of the experiments with different solute concentrations are shown in Fig. 9. The two figures indicate that both the degree of superheating and the relative superheating are higher in liquids with low solute concentration. However, it is difficult to compare the presented results, because not only the solute concentration was different in these experiments, but also the average crystal size. As shown above, the average crystal size influences superheating significantly. A more comprehensive analysis is therefore presented in the next section to identify the influence of the solute concentration on superheating.
Fig. 7 – Degree of superheating (a) and relative superheating z (b) at the outlet for various ice slurry velocities (experiments 1–4).
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Fig. 8 – Degree of superheating (a) and relative superheating z (b) at the outlet for various heat fluxes (TEG,in) and crystal sizes (experiments 8–10).
5.
Discussion
The presented results for superheating at the outlet of the heat exchanger can be explained by a model of the melting process. This model is based on the heat and mass transfer processes in a control volume of the heat exchanger as shown in Fig. 10. It is assumed that the control volume is ideally mixed, which means that ice slurry is homogeneously distributed and that the liquid temperature is constant in the entire control volume. The control volume is considered as a steady state system and the heat balance is therefore: _ is his;out his;in (16) Q_ ¼ m
The heat transferred from the wall to the liquid in the control volume is given by Q_ ¼ ai Ai Tw Tliq (17) with Ai ¼ pdi;inner Dx The increase of the enthalpy of ice slurry in Eq. (16) is represented by his;out his;in ¼ ð1 fout Þhliq;out þ fout hice;out ð1 fin Þhliq;in þ fin hice;in (18) The ice fraction at the outlet in Eq. (18) can be replaced by fout ¼ fin Df
(19)
Fig. 9 – Degree of superheating (a) and relative superheating z (b) at the outlet for various solute concentrations (experiments 2, 6 and 7).
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Aice ¼ c1 Nd2Feret
(26)
It is assumed here that both the shape of the individual crystals and the shape of the crystal size distribution were the same in the various experiments. The number of crystals N in Eq. (26) is deduced from the total mass of ice in the control volume using the same assumptions: N¼
Fig. 10 – Schematic representation of melting process in a control volume.
Combining Eqs. (18) and (19) gives the following expression for the change in enthalpy: his;out his;in ¼ Df hliq;out hice;out þ ð1 fin Þ hliq;out hliq;in þ fin hice;out hice;in
(20)
The change in liquid enthalpy is approximated by the product of the temperature increase and the specific heat of the liquid. It is assumed here that the heat of mixing can be neglected and that specific heats are constant for small temperature changes. With these assumptions, Eq. (20) becomes his;out his;in zDf Dhf þ DT ð1 fin Þcp;liq þ fin cp;ice
(21)
The enthalpy increase consists of a latent heat contribution represented by a decrease of the ice fraction and a sensible heat contribution represented by an increase of the temperature. During the initial stage of the melting experiments, the sensible heat contribution was 20% of the total enthalpy increase on average. For simplicity, the sensible heat contribution is neglected in this analysis and the enthalpy difference is assumed equal to the product of the change in ice fraction and the latent heat of fusion: his;out his;in zDf Dhf
(22)
Combining Eqs. (16), (17) and (22) leads to the following heat balance for the control volume: _ is Df Dhf ai pdi;inner Dx Tw Tliq ¼ m
mice p with mice ¼ fmis ¼ fris d2i;inner Dx 4 c2 rice d3Feret
(27)
The negative crystal growth rate G in Eq. (25) is determined by mass and heat transfer between the crystal surface and the liquid phase of the slurry (Mersmann et al., 2001): G¼
T Tliq BA eq vTeq Dhf 3rice BV w þ rliq k vw acr
(28)
Rearranging Eq. (28) explicitly shows the ratio between the heat and mass transfer coefficient: G¼
Teq Tliq BA ! 3rice BV Dh acr w vTeq f þ 1 acr k Dhf r vw liq
(29)
This ratio of the coefficients is determined from the analogy between heat and mass transfer close to the crystal surface (Holman, 1997): 1=3 1=3 2=3 acr Nucr lliq Pr1=3 lliq cp;liq rliq lliq ¼ ¼ 1=3 ¼ k Shcr D Sc D D2=3
(30)
Eq. (30) can then be substituted in Eq. (29) resulting in a new expression for the crystal growth rate: G¼
BA acr 3rice BV Dhf
Teq Tliq ! 1=3 2=3 cp;liq lliq w vTeq þ 1 2=3 vw D2=3 Dhf rliq
(31)
Eq. (31) shows that both heat and mass transfer resistances determine the total resistance for melting. However, the ratio of these contributions strongly depends on the solute concentration as shown in Fig. 11. At low solute concentration, 3.5 wt% for example, the crystal growth rate is determined by equal contributions of heat and mass transfer resistance,
(23)
The decrease of the ice fraction is caused by the melting of individual ice crystals. The mass reduction of ice in the control volume is proportional to the total surface of ice crystals Aice and the negative growth rate G: _ ice ¼ rice Aice G Dm
(24)
The decrease of the ice fraction is now calculated as the ratio between the reduction of the ice mass and the mass flow rate of ice slurry: Df ¼
_ ice Dm r Aice G ¼ ice _ is _ is m m
(25)
The total available crystal surface Aice for the melting process is proportional to the number of crystals in the control volume and the characteristic crystal size squared:
Fig. 11 – Contributions to crystal growth resistance relative to heat transfer resistance.
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while at high concentrations the growth rate is almost completely determined by mass transfer. Eqs. (26), (27) and (31) are now substituted in Eq. (25): Df ¼
c1 p BA ris fd2i;inner Dx acr _ is Dhf c2 12 BV rice dFeret m
Tliq Teq 1=3 2=3 cp;liq lliq w vTeq 2=3 vw D2=3 Dhf rliq
! þ1 (32)
Substitution of Eq. (32) in the heat balance of Eq. (23) finally gives an expression for the degree of superheating: c2 12BV rice dFeret 1 ai DTsh ¼ Tliq Teq ¼ c1 BA ris di;inner f acr 1 0 1=3 2=3 cp;liq lliq w vT eq þ 1A Tw Tliq @ 2=3 2=3 vw D Dhf rliq
ð33Þ
Eq. (33) shows that the degree of superheating is higher for slurries with large crystals, which is in accordance with the experiments. Ice slurries with large crystals have a relatively small crystal surface resulting in a slow melting process and exhibit therefore high degrees of superheating. Ice slurries with low ice fractions have also relatively little crystal surface and exhibit therefore also high degrees of superheating. This phenomenon is represented in Eq. (33) by the ice fraction in the denominator. In correspondence with the experiments, Eq. (33) shows that the degree of superheating increases with increasing heat flux, which is represented here by the temperature difference between wall and liquid. However, the ratio between the driving forces of the two stages of melting is not influenced by the heat flux. Therefore, the relative superheating does not depend on the heat flux, which is in accordance with the experiments (see Fig. 8). The experimental results show that the relative superheating is hardly influenced by the ice slurry velocity. This observation can also be explained by Eq. (33). A higher ice slurry velocity results first of all in a higher heat transfer coefficient between wall and liquid. However, the rates of the heat and mass transfer processes between crystals and liquid also increase. It is expected that the relative increases of all these coefficients are approximately similar as the velocity increases and that therefore the relative superheating is almost independent of the ice slurry velocity. According to Eq. (33), the degree of superheating is higher in aqueous solutions with higher solute concentrations. This trend cannot directly be confirmed by the experiments, because the experiments with different solute concentrations also had different average crystals sizes. In order to confirm the influence of the solute concentration, all variables that have been varied in the experiments have been considered simultaneously. For this purpose, all experimental constants of Eq. (33) are combined in one constant c3:
DTsh
1 0 1=3 2=3 vTeq rice dFeret @ cp;liq lliq w þ 1A Tw Tliq ¼ c3 ris f vw D2=3 Dhf r2=3 liq
c2 12BV ai with c3 ¼ c1 BA di;inner acr
ð34Þ
The ratio of the heat transfer coefficients in the expression for c3 is assumed constant here. The experiments with
Fig. 12 – Relation between variables at right-hand side of Eq. (34) and measured degrees of superheating; the numbers in the figure represent the experiment number as listed in Table 1.
different ice slurry velocities showed similar relative superheating values indicating that this assumption is reasonable. The experimental variables at the right-hand side of Eq. (34) are considered at the start of each experiment. This analysis is limited to the initial phase of the experiments, since the average ice crystal size was only determined prior to each experiment. It is expected that the average crystal size decreases in the course of an experiment, but this was not quantified. The results of this analysis for all the 10 melting experiments shown in Fig. 12 confirm proportionality between the variables and the degree of superheating stated in Eq. (34). The expression in Eq. (33) shows that the degree of superheating also depends on the tube diameter. According to the expression, the degree of superheating decreases with increasing tube diameter. Since the diameter of the heat exchanger tube was not varied in the experiments, this influence cannot be confirmed.
6.
Conclusions
The liquid temperature of ice slurry in melting heat exchangers can be significantly higher than the equilibrium temperature. This phenomenon is referred to as superheating and can lead to a serious reduction of heat exchanger capacities. The degree of superheating at the outlet of heat exchangers is proportional to the average ice crystal size and the heat flux. It is furthermore inversely proportional to the ice fraction and increases therefore as the ice fraction decreases. The negative growth rate of melting ice crystals is determined by both heat and mass transfer. The degree of superheating increases therefore with increasing solute concentration, especially at higher concentrations at which the heat transfer resistance plays a minor role. Finally, the degree of superheating is expected to be higher in heat exchangers with small hydraulic diameters.
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references
Ayel, V., Lottin, O., Peerhossaini, H., 2003. Rheology, flow behaviour and heat transfer of ice slurries: a review of the state of the art. Int. J. Refrigeration 26, 95–107. Egolf, P.W., Kitanovski, A., Ata-Caesar, D., Stamatiou, E., Kawaji, M., Bedecarrats, J.P., Strub, F., 2005. Thermodynamics and heat transfer of ice slurries. Int. J. Refrigeration 28, 51–59. Fikiin, K., Wang, M.J., Kauffeld, M., Hansen, T.M., 2005. Direct contact chilling and freezing of foods in ice slurries. In: Kauffeld, M., Kawaji, M., Egolf, P.W. (Eds.), Handbook on Ice Slurries: Fundamentals and Engineering. International Institute of Refrigeration, Paris, pp. 251–271. Frei, B., Boyman, T., 2003. Plate heat exchanger operating with ice slurry. In: Proceedings of the 1st International Conference and Business Forum on Phase Change Materials and Phase Change Slurries, 2003 April 23–26, Yverdonles-Bains, Switzerland. Hansen, T.M., Radosˇevic´, M., Kauffeld, M., Zwieg, T., 2003. Investigation of ice crystal growth and geometrical characteristics in ice slurry. HVAC&R Res. 9, 9–32. Holman, J.P., 1997. Heat Transfer, Eighth ed. McGraw-Hill, New York. Kitanovski, A., Vuarnoz, D., Ata-Caesar, D., Egolf, P.W., Hansen, T. M., Doetsch, C., 2005. The fluid dynamics of ice slurry. Int. J. Refrigeration 28, 37–50.
Lee, D.W., Yoon, E.S., Joo, M.C., Sharma, A., 2006. Heat transfer characteristics of the ice slurry at melting process in a tube flow. Int. J. Refrigeration 29, 451–455. Mersmann, A., Eble, A., Heyer, C., 2001. Crystal growth. In: Mersmann, A. (Ed.), Crystallization Technology Handbook, second ed. Marcel Dekker, Inc, New York, pp. 81–143. _ Niezgoda-Zelasko, B., 2006. Heat transfer of ice slurry flows in tubes. Int. J. Refrigeration 29, 437–450. _ Niezgoda-Zelasko, B., Zalewski, W., 2006. Momentum transfer of ice slurry flows in tubes, experimental investigations. Int. J. Refrigeration 29, 418–428. Pronk, P., Hansen, T.M., Infante Ferreira, C.A., Witkamp, G.J., 2005a. Time-dependent behavior of different ice slurries during storage. Int. J. Refrigeration 28, 27–36. Pronk, P., Infante Ferreira, C.A., Witkamp, G.J., 2005b. A dynamic model of Ostwald ripening in ice suspensions. J. Cryst. Growth 275, e1355–e1361. Pronk, P., Infante Ferreira, C.A., Witkamp, G.J., 2006. Influence of solute type and concentration on ice scaling in fluidized bed ice crystallizers. Chem. Eng. Sci. 61, 4354–4362. Stamatiou, E., Kawaji, M., 2005. Thermal and flow behavior of ice slurries in a vertical rectangular channel – part II: Forced convective melting heat transfer. Int. J. Heat Mass Transfer 48, 3544–3559. Torres-de Marı´a, G., Abril, J., Casp, A., 2005. Coefficients d’e´changes superficiels pour la re´frige´ration et la conge´lation d’aliments immerge´s dans un coulis de glace. Int. J. Refrigeration 28, 1040–1047.
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Superheating of ice slurry in melting heat exchangers P. Pronk*, C.A. Infante Ferreira, G.J. Witkamp Process and Energy Department, Delft University of Technology, Leeghwaterstraat 44, 2628 CA, Delft, The Netherlands
article info
abstract
Article history:
One of the main components of an ice slurry system is the melting heat exchanger, in
Received 24 September 2006
which ice slurry absorbs heat resulting in the melting of ice crystals. Design calculations
Received in revised form
of melting heat exchangers are mainly based on heat transfer and pressure drop data,
4 September 2007
but recent experimental studies have shown that superheating of ice slurry should also
Accepted 16 September 2007
be considered. This paper presents ice slurry melting experiments with a tube-in-tube
Published online 22 November 2007
heat transfer coil. The experimental results indicate that operating conditions such as ice slurry velocity, heat flux, solute concentration, ice fraction, and ice crystal size deter-
Keywords:
mine the degree of superheating. The various influences are explained by considering
Ice slurry
the melting process as a two-stage process consisting of the heat transfer between wall
Heat exchanger
and liquid and the combined heat and mass transfer between liquid and crystals. Bigger
Melting
ice crystals and higher solute concentrations decrease the rate of the second stage and
Experiment
therefore increase the degree of superheating. ª 2007 Elsevier Ltd and IIR. All rights reserved.
Parameter Heat transfer Mass transfer Superheating
Surchauffe du coulis de glace dans les e´changeurs de chaleur avec fonte de glace Mots cle´s : Coulis de glace ; E´changeur de chaleur ; Fusion ; Expe´rimentation ; Parame`tre ; Transfert de chaleur ; Transfert de masse ; Surchauffe
1.
Introduction
In the last 10 years, ice slurry has successfully been applied as secondary refrigerant in refrigeration and air-conditioning installations. The application of ice slurry minimizes the required amount of primary refrigerant and enables thermal storage. The latter makes it possible to produce ice slurry at
night with the benefits of low electricity tariffs and low condensing pressures, and use it in daytime when cooling loads normally peak. Proper application of thermal storage leads to a reduction of investment and/or energy costs. After production and storage, ice slurry is transported to applications where it melts and provides cooling for rooms, products or processes. In general, two different methods of ice
* Corresponding author. Tel.: þ31 15 2784894; fax: þ31 15 2786975. E-mail address: [email protected] (P. Pronk). 0140-7007/$ – see front matter ª 2007 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2007.09.008
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Nomenclature A BA BV c1.3 cp dFeret d D G h Dhf k m _ m N n Nucr Dp Pr Q_ Sc Shcr T DTln DTsh t Dt U u V_ w
area (m2) area shape factor volume shape factor constants specific heat (J/kg K) average crystal Feret diameter (m) tube diameter (m) diffusion coefficient (m2/s) growth rate (m/s) enthalpy (J/kg) latent heat of fusion of ice (J/kg) mass transfer coefficient (m/s) mass (kg) mass flow (kg/s) number of crystals number of measurements crystal Nusselt number, acrdFeret/lliq pressure drop (Pa) Prandtl number, cp,liqmliq/lliq heat (W) Schmidt number, mliq/(rliqD) crystal Sherwood number, kdFeret/D temperature ( C) logarithmic mean temperature difference (K) degree of superheating (K), defined in Eq. (11) time (s) measurement interval (s) overall heat transfer coefficient (W/m2 K) velocity (m/s) volume flow rate (m3/s) solute mass fraction
slurry melting are used. In the first method, ice slurry is directly poured onto food products for example fish, resulting in high cooling rates and therefore high product qualities (Fikiin et al., 2005; Torres-de Marı´a et al., 2005). In the second method, ice slurry absorbs heat in regular heat exchangers cooling a second fluid, for example air in display cabinets for supermarkets or air-conditioning systems. Design calculations for this kind of heat exchangers are mainly based on experimentally obtained heat transfer and pressure drop data for melting ice slurry (Ayel et al., 2003; Egolf et al., 2005; Lee et al., 2006; Niez_ _ goda-Zelasko, 2006; Niezgoda-Zelasko and Zalewski, 2006; Stamatiou and Kawaji, 2005). Recent experimental studies have shown that ice slurry may be significantly superheated at the outlet of melting heat exchangers (Hansen et al., 2003; Kitanovski et al., 2005; Frei and Boyman, 2003). This phenomenon should also be taken into account in design calculations, since superheating increases the average ice slurry temperature in the heat exchanger to a value that is higher than expected from equilibrium calculations. As a result, the temperature difference between ice slurry and the other fluid is lower resulting in reduced heat exchanger capacities. Ice slurry is considered superheated when its liquid temperature is higher than its equilibrium temperature.
w0 Dx
initial solute mass fraction in liquid length of control volume
Greek a z l m r s f
heat transfer coefficient (W/m2 K) relative superheating, defined in Eq. (12) thermal conductivity (W/m K) viscosity (Pa s) density (kg/m3) period (h) ice mass fraction
Subscripts cr crystal EG ethylene glycol solution eq equilibrium eqin equilibrium at inlet fr freeze point he heat exchanger i inside ice ice in inlet init initial inner inner is ice slurry liq liquid meas measured o outside out outlet real real rest other components stor storage w wall
Superheating can be explained by considering the melting process of ice slurry as a two-stage process. First the heat exchanger wall heats the liquid and subsequently the superheated liquid melts the ice crystals. The relation between the rates of both processes determines the degree of superheating. For example, when crystal-to-liquid heat and mass transfer processes are relatively slow compared to the wallto-liquid heat transfer process, the degree of superheating is high. The degree of superheating is expected to depend on operating conditions, such as ice slurry velocity, heat flux, solute concentration, ice fraction and ice crystal size. However, quantitative data on superheating are only reported by Frei and Boyman (2003) indicating that the degree of superheating increases as the ice fraction decreases. Moreover, the physical mechanisms that determine the degree of superheating are not understood yet. The aims of this paper are therefore to investigate the influence of operating conditions on the degree of superheating and to unravel the physical mechanisms behind this phenomenon. For this purpose, experiments on ice slurry melting have been performed in a tube-in-tube heat transfer coil, during which the influences of various operating conditions on superheating have been investigated.
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2.
Experimental method
Table 1 – Experimental conditions of melting experiments
2.1.
Experimental set-up
Exp. no.
The melting experiments were performed with the experimental set-up shown in Fig. 1. A fluidized bed heat exchanger as described by Pronk et al. (2006) was used to produce ice slurry from aqueous sodium chloride solutions. The produced ice slurry was stored in an insulated tank that was equipped with a mixing device to keep the ice slurry homogeneous. The tank could easily be disconnected from the set-up and be placed in a cold room for isothermal storage. After production and eventually isothermal storage, the ice crystals were analyzed with a visualization section consisting of a flow cell and a microscope. The ice slurry was subsequently pumped through the inner tube of a tube-in-tube heat transfer coil, which had an internal diameter of 7.0 mm, an outside diameter of 9.5 mm and a total external heat-exchanging surface of 0.181 m2. A 20 wt% ethylene glycol solution extracted from a thermostatic bath flowed counter currently through the annulus with a hydraulic diameter of 6.2 mm and heated the ice slurry in the inner tube. The melting process was continued until all ice crystals were molten and the tank contained only liquid. PT-100 elements with an accuracy of 0.01 K measured the temperatures at the inlets and outlets of the heat exchanger. The mass flow of ice slurry was measured using a coriolis mass flow meter and a magnetic flow meter was used to measure the flow rate of the ethylene glycol solution. The coriolis mass flow meter was also able to measure the temperature of ice slurry downstream of the heat exchanger. All flow rates and temperatures were automatically measured every 10 s, with the exception of the temperature measured in the coriolis mass flow meter which was manually recorded.
2.2.
Experimental conditions
The experimental series consisted of 10 melting experiments, in which the operating conditions were systematically varied as shown in Table 1. The varied conditions were the ice slurry velocity, the heat flux, the ice crystal size and the sodium chloride concentration. The heat flux was adjusted by varying the
1 2 3 4 5 6 7 8 9 10
Tfr ( C)
uis (m/s)
TEG,in ( C)
sstor (h)
6.6 6.6 6.6 6.6 6.6 3.5 11.0 7.1 7.0 7.1
4.1 4.1 4.1 4.1 4.1 2.1 7.4 4.4 4.4 4.4
1.0 1.5 2.0 2.5 1.5 1.5 1.5 1.8 1.7 1.8
3.0 3.0 3.0 3.0 3.0 5.2 0.7 2.6 2.6 0.0
0 0 0 0 16 0 0 0 15 0
dFeret (mm) 249a 249 249a 249a 283 338 133 148 277 148b
fin;init (wt%) 17 18 18 16 16 14 17 10 9 10
a Assumed equal as in experiment 2. b Assumed equal as in experiment 8.
inlet temperature of the aqueous ethylene glycol solution. In most experiments, the difference between the initial freezing temperature of the aqueous solution and the inlet temperature of the ethylene glycol solution was 7.1 0.2 K, except for experiment 10 where this temperature difference was only 4.4 K. In this last experiment the heat flux varied from 4 to 7 kW/m2, while the heat flux in the other experiments was from 7 to 13 kW/m2. The average crystal size was determined by analyzing the crystals with a visualization section. In this respect, the Feret diameter was used as characteristic crystal size, which is defined as the diameter of a circle with the same area as the projection of the crystal (Pronk et al., 2005a). Ice crystals produced from aqueous solutions with equal solute concentration and equal production procedure appeared to have approximately the same average crystal size. The average crystal sizes at the start of experiments 1, 3 and 4 were therefore assumed equal to the average crystal size determined at the start of experiment 2. The same assumption was made for the crystal sizes of experiments 8 and 10. Ice crystals produced from aqueous solutions with higher solute concentrations appeared to have smaller crystals. In order to vary the average crystal size for a certain solute concentration, ice slurry was isothermally stored in the cold room. During isothermal storage, the average crystal size increased as a result of Ostwald ripening (Pronk et al., 2005a,b). In general, the 10 melting experiments listed in Table 1 all showed the same trends on superheating. Therefore the observed phenomena are discussed for one experiment only, namely experiment 1. Subsequently, the results of the different experiments are compared.
3.
Fig. 1 – Schematic overview of the experimental set-up.
w0 (wt%)
Results: analysis of a single experiment
At the start of experiment 1, the ice fraction at the inlet was 17 wt% while the temperature was 5.0 C (see Fig. 2). According to the heat balance, the reduction in ice fraction was initially approximately 9 wt% per pass, which resulted in an expected outlet ice fraction of about 8 wt% and an expected outlet temperature below the initial freezing temperature of 4.1 C. However, the measured outlet temperature exceeded this initial freezing temperature by about 1 K, which means
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Fig. 2 – Measured ice slurry temperatures and ice fraction at the inlet based on equilibrium during melting experiment 1.
that the ice slurry at the outlet was superheated. The temperature measured in the coriolis mass flow meter was below the measured outlet temperature, which is attributed to the release of superheating downstream of the heat exchanger.
3.1.
Assuming equilibrium at the inlet
If it assumed that ice slurry entering the heat exchanger is in equilibrium, then the ice fraction at the inlet is calculated by fin;eqin ¼
win;eqin w0 win;eqin
(1)
In Eq. (1), the weight fraction of solute in the solution win,eqin is determined from the measured inlet temperature assuming equilibrium: (2) win;eqin ¼ weq Tin;meas The development of the inlet ice fraction fin,eqin shown in Fig. 2 suggests that all ice crystals had melted at t ¼ 2400 s. However, by that time ice crystals were still observed in the ice suspension tank. Another indication that ice crystals were still present in the system is that the rate of change of the measured inlet temperature does not change significantly at t ¼ 2400 s. However, a considerable difference is observed at t ¼ 3200 s indicating that all ice crystals had melted by that time. The described observations indicate that ice slurry is also not in equilibrium at the inlet of the heat exchanger, at least during the final stage of the experiment. For a correct analysis, ice slurry should therefore be considered as a non-equilibrium fluid at both inlet and outlet.
3.2.
his;in;eqin ¼ 1 fin;eqin hliq win;eqin ; Tin;meas for Tin;meas Tfr þ fin;eqin hice Tin;meas
ð3Þ
his;in;eqin ¼ hliq w0 ; Tin;meas for Tin;meas > Tfr
(4)
For temperatures above the freezing temperature, the enthalpy simply equals the enthalpy of the aqueous solution. At temperatures below the freezing temperature, the enthalpy of ice slurry is the weighted average of the enthalpy of the solution and the enthalpy of ice. The enthalpy based on equilibrium calculations can be compared with the enthalpy based on the cumulative heat input, which consists of the heat transferred in the heat exchanger and the heat input by other components, such as the pump and the mixing device in the tank: Z t _ Q he þ Q_ rest Þ vt (5) his;in;real ftg ¼ his;in;eqin ft ¼ 0g þ mis 0 The enthalpy at t ¼ 0 is determined by assuming that ice slurry is in equilibrium at the beginning of the experiment. The integral in Eq. (5) is rewritten into a summation in order to apply it to the measured data: his;in;real ftg ¼ his;in;eqin ft ¼ 0g þ
n¼t=Dt X n¼0
Q_ he þ Q_ rest ÞDt mis
The heat input by the other components is determined by equating the enthalpy based on equilibrium with the enthalpy based on the temperature measurement for the final measurement at t ¼ 3400 s (see Fig. 3). By this time, no ice crystals were present in the tank and therefore the enthalpy based on the temperature represents the real enthalpy. Application of this method to melting experiment 1 results in a heat input by the other components of 170 W. Heat input values calculated for the other experiments showed similar numbers. Since the real enthalpy at the inlet of the heat exchanger is known from Eq. (6), the enthalpy at the outlet can be calculated by his;out;real ¼ his;in;real þ
Q_ is _ is m
Assuming non-equilibrium at the inlet
In order to quantify superheating at both the inlet and outlet of the heat exchanger, the enthalpy of ice slurry at both locations is considered. First, the enthalpy at the inlet is considered for the assumption of equilibrium at this location:
(6)
Fig. 3 – Ice slurry enthalpies at the inlet during melting experiment 1.
(7)
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When ice slurry is not in equilibrium, the ice fraction cannot be calculated by using the initial solute concentration w0 and the measured temperature only (see Eqs. (3) and (4)). The nonequilibrium state requires a third thermodynamic property to calculate the ice fraction, for example enthalpy. The enthalpy of ice slurry that is not in equilibrium is given by: (8) his;real ¼ ð1 freal Þhliq fwreal ; Tmeas g þ freal hice Teq fwreal g At the inlet and outlet of the heat exchanger, the enthalpies are known from Eqs. (6) and (7) and the temperature is known from measurements. Thus Eq. (8) contains only two unknown variables, namely the solute concentration in the solution wreal and the ice fraction freal. Since the ice slurry is homogeneously mixed in the tank, it is assumed that the solute concentration in the slurry always equals the initial solute concentration w0. The ice fraction freal is therefore directly related to the solute concentration in the solution wreal by means of the solute mass balance: w0 ¼ ð1 freal Þwreal
(9)
The ice fractions at the inlet and outlet can now be solved iteratively from Eqs. (8) and (9), which is shown for experiment 1 in Fig. 4. The solute concentration in the liquid wreal is subsequently used to calculate the equilibrium temperature at the inlet and outlet of the heat exchanger: Teq ¼ Teq fwreal g
(10)
Fig. 5 shows that the measured outlet temperatures exceed the calculated equilibrium temperatures at the outlet indicating that the ice slurry is considerably superheated at this location.
3.3.
Superheating definition
In order to quantify superheating at the outlet of the heat exchanger, the degree of superheating DTsh is defined as the difference between the measured and the equilibrium temperature: DTsh ¼ Tliq;meas Teq fwreal g
(11)
The melting of ice slurry in a heat exchanger can be considered as a process consisting of two stages as shown in Fig. 6.
Fig. 5 – Measured ice slurry temperature, and calculated ice fractions and equilibrium temperatures at the outlet of the heat exchanger during melting experiment 1.
The first stage consists of the heat transfer process from the wall to the liquid. The driving force for this process is the temperature difference between the wall and the liquid phase. The second stage is the actual melting of the ice crystals, where the difference between the liquid temperature and the equilibrium temperature, hence the degree of superheating, is the driving force. The degree of superheating can therefore be seen as a fraction of the total driving force of the melting process: z¼
Tmeas Teq DTsh ¼ DTw-liq þ DTsh Tw Teq
(12)
This relative superheating z enables the comparison of the superheating results from experiments with different mass flow rates and different heat fluxes. For the analysis of the relative superheating, it is necessary to calculate the wall temperature at the ice slurry side of the inner tube. Here, the ratio of heat transfer coefficients, the ice slurry temperature and the temperature of the ethylene glycol solution are used to determine this temperature: ðTw Tis Þ Uo do;inner ¼ ðTEG Tis Þ ai di;inner
(13)
The overall heat transfer coefficient Uo is deduced from the total heat flux in the heat exchanger, which was determined
Fig. 4 – Measured ice slurry temperatures and calculated ice fractions during melting experiment 1.
Fig. 6 – Schematic representation of temperatures during melting of ice slurry in a heat exchanger.
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from the flow rate and the inlet and outlet temperatures of the ethylene glycol solution: Q_ he ¼ Uo Ao DTln
(14)
The logarithmic temperature difference in Eq. (14) is calculated from the measured temperatures at the inlets and outlets of the heat exchanger. The overall heat transfer resistance (1/Uo) consists of three parts, the annular side heat transfer resistance, the heat transfer resistance of the tube wall, and the wall-to-liquid heat transfer resistance: 1 1 do;inner do;inner 1 do;inner ¼ þ ln (15) þ Uo ao ai di;inner 2lw di;inner The Wilson plot calibration technique was used to formulate single-phase heat transfer correlations for both sides of the heat exchanger. The correlation for the annular side was used to calculate the heat transfer coefficient of the ethylene glycol flow ao. Subsequently, this heat transfer coefficient is used to determine the heat transfer coefficient for the ice slurry flow ai from Eq. (15). Finally, both the overall heat transfer coefficient and the ice slurry heat transfer coefficient are used to calculate the wall temperature from Eq. (13).
4.
Results: influence of operating conditions
4.1.
Influence of ice fraction and ice slurry velocity
the relative superheating of the experiments with slurry velocities of 1.0, 1.5 and 2.0 m/s are very similar, but that the relative superheating at a velocity of 2.5 m/s is slightly lower.
4.2.
The results from the experiments with different ethylene glycol solution inlet temperatures in Fig. 8a show that the degree of superheating increases as the heat flux increases. However, the relative superheating z is similar for different heat fluxes as shown in Fig. 8b.
4.3.
Influence of crystal size
The results of experiments 8 and 10 in Fig. 8a indicate that ice slurries consisting of larger crystals exhibit higher degrees of superheating. Accordingly, the relative superheating also increases as the average ice crystal size increases (see Fig. 8b). A comparison of the superheating results of experiments 2 and 5, in which the crystal size was also the only varied variable, leads to the same conclusion.
4.4.
The superheating results for different ice slurry velocities in Fig. 7a clearly show that the degree of superheating increases as the ice fraction decreases. The figure also shows that for ice fractions higher than 5 wt%, the degree of superheating is higher in the experiments with low ice slurry velocities. This higher degree of superheating is mainly the result of the higher wall temperature caused by the lower wall-toliquid heat transfer coefficient at low slurry velocities. The results for the relative superheating z in Fig. 7b take these different wall temperatures into account. This figure shows that
Influence of heat flux
Influence of solute concentration
The superheating results of the experiments with different solute concentrations are shown in Fig. 9. The two figures indicate that both the degree of superheating and the relative superheating are higher in liquids with low solute concentration. However, it is difficult to compare the presented results, because not only the solute concentration was different in these experiments, but also the average crystal size. As shown above, the average crystal size influences superheating significantly. A more comprehensive analysis is therefore presented in the next section to identify the influence of the solute concentration on superheating.
Fig. 7 – Degree of superheating (a) and relative superheating z (b) at the outlet for various ice slurry velocities (experiments 1–4).
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Fig. 8 – Degree of superheating (a) and relative superheating z (b) at the outlet for various heat fluxes (TEG,in) and crystal sizes (experiments 8–10).
5.
Discussion
The presented results for superheating at the outlet of the heat exchanger can be explained by a model of the melting process. This model is based on the heat and mass transfer processes in a control volume of the heat exchanger as shown in Fig. 10. It is assumed that the control volume is ideally mixed, which means that ice slurry is homogeneously distributed and that the liquid temperature is constant in the entire control volume. The control volume is considered as a steady state system and the heat balance is therefore: _ is his;out his;in (16) Q_ ¼ m
The heat transferred from the wall to the liquid in the control volume is given by Q_ ¼ ai Ai Tw Tliq (17) with Ai ¼ pdi;inner Dx The increase of the enthalpy of ice slurry in Eq. (16) is represented by his;out his;in ¼ ð1 fout Þhliq;out þ fout hice;out ð1 fin Þhliq;in þ fin hice;in (18) The ice fraction at the outlet in Eq. (18) can be replaced by fout ¼ fin Df
(19)
Fig. 9 – Degree of superheating (a) and relative superheating z (b) at the outlet for various solute concentrations (experiments 2, 6 and 7).
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Aice ¼ c1 Nd2Feret
(26)
It is assumed here that both the shape of the individual crystals and the shape of the crystal size distribution were the same in the various experiments. The number of crystals N in Eq. (26) is deduced from the total mass of ice in the control volume using the same assumptions: N¼
Fig. 10 – Schematic representation of melting process in a control volume.
Combining Eqs. (18) and (19) gives the following expression for the change in enthalpy: his;out his;in ¼ Df hliq;out hice;out þ ð1 fin Þ hliq;out hliq;in þ fin hice;out hice;in
(20)
The change in liquid enthalpy is approximated by the product of the temperature increase and the specific heat of the liquid. It is assumed here that the heat of mixing can be neglected and that specific heats are constant for small temperature changes. With these assumptions, Eq. (20) becomes his;out his;in zDf Dhf þ DT ð1 fin Þcp;liq þ fin cp;ice
(21)
The enthalpy increase consists of a latent heat contribution represented by a decrease of the ice fraction and a sensible heat contribution represented by an increase of the temperature. During the initial stage of the melting experiments, the sensible heat contribution was 20% of the total enthalpy increase on average. For simplicity, the sensible heat contribution is neglected in this analysis and the enthalpy difference is assumed equal to the product of the change in ice fraction and the latent heat of fusion: his;out his;in zDf Dhf
(22)
Combining Eqs. (16), (17) and (22) leads to the following heat balance for the control volume: _ is Df Dhf ai pdi;inner Dx Tw Tliq ¼ m
mice p with mice ¼ fmis ¼ fris d2i;inner Dx 4 c2 rice d3Feret
(27)
The negative crystal growth rate G in Eq. (25) is determined by mass and heat transfer between the crystal surface and the liquid phase of the slurry (Mersmann et al., 2001): G¼
T Tliq BA eq vTeq Dhf 3rice BV w þ rliq k vw acr
(28)
Rearranging Eq. (28) explicitly shows the ratio between the heat and mass transfer coefficient: G¼
Teq Tliq BA ! 3rice BV Dh acr w vTeq f þ 1 acr k Dhf r vw liq
(29)
This ratio of the coefficients is determined from the analogy between heat and mass transfer close to the crystal surface (Holman, 1997): 1=3 1=3 2=3 acr Nucr lliq Pr1=3 lliq cp;liq rliq lliq ¼ ¼ 1=3 ¼ k Shcr D Sc D D2=3
(30)
Eq. (30) can then be substituted in Eq. (29) resulting in a new expression for the crystal growth rate: G¼
BA acr 3rice BV Dhf
Teq Tliq ! 1=3 2=3 cp;liq lliq w vTeq þ 1 2=3 vw D2=3 Dhf rliq
(31)
Eq. (31) shows that both heat and mass transfer resistances determine the total resistance for melting. However, the ratio of these contributions strongly depends on the solute concentration as shown in Fig. 11. At low solute concentration, 3.5 wt% for example, the crystal growth rate is determined by equal contributions of heat and mass transfer resistance,
(23)
The decrease of the ice fraction is caused by the melting of individual ice crystals. The mass reduction of ice in the control volume is proportional to the total surface of ice crystals Aice and the negative growth rate G: _ ice ¼ rice Aice G Dm
(24)
The decrease of the ice fraction is now calculated as the ratio between the reduction of the ice mass and the mass flow rate of ice slurry: Df ¼
_ ice Dm r Aice G ¼ ice _ is _ is m m
(25)
The total available crystal surface Aice for the melting process is proportional to the number of crystals in the control volume and the characteristic crystal size squared:
Fig. 11 – Contributions to crystal growth resistance relative to heat transfer resistance.
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while at high concentrations the growth rate is almost completely determined by mass transfer. Eqs. (26), (27) and (31) are now substituted in Eq. (25): Df ¼
c1 p BA ris fd2i;inner Dx acr _ is Dhf c2 12 BV rice dFeret m
Tliq Teq 1=3 2=3 cp;liq lliq w vTeq 2=3 vw D2=3 Dhf rliq
! þ1 (32)
Substitution of Eq. (32) in the heat balance of Eq. (23) finally gives an expression for the degree of superheating: c2 12BV rice dFeret 1 ai DTsh ¼ Tliq Teq ¼ c1 BA ris di;inner f acr 1 0 1=3 2=3 cp;liq lliq w vT eq þ 1A Tw Tliq @ 2=3 2=3 vw D Dhf rliq
ð33Þ
Eq. (33) shows that the degree of superheating is higher for slurries with large crystals, which is in accordance with the experiments. Ice slurries with large crystals have a relatively small crystal surface resulting in a slow melting process and exhibit therefore high degrees of superheating. Ice slurries with low ice fractions have also relatively little crystal surface and exhibit therefore also high degrees of superheating. This phenomenon is represented in Eq. (33) by the ice fraction in the denominator. In correspondence with the experiments, Eq. (33) shows that the degree of superheating increases with increasing heat flux, which is represented here by the temperature difference between wall and liquid. However, the ratio between the driving forces of the two stages of melting is not influenced by the heat flux. Therefore, the relative superheating does not depend on the heat flux, which is in accordance with the experiments (see Fig. 8). The experimental results show that the relative superheating is hardly influenced by the ice slurry velocity. This observation can also be explained by Eq. (33). A higher ice slurry velocity results first of all in a higher heat transfer coefficient between wall and liquid. However, the rates of the heat and mass transfer processes between crystals and liquid also increase. It is expected that the relative increases of all these coefficients are approximately similar as the velocity increases and that therefore the relative superheating is almost independent of the ice slurry velocity. According to Eq. (33), the degree of superheating is higher in aqueous solutions with higher solute concentrations. This trend cannot directly be confirmed by the experiments, because the experiments with different solute concentrations also had different average crystals sizes. In order to confirm the influence of the solute concentration, all variables that have been varied in the experiments have been considered simultaneously. For this purpose, all experimental constants of Eq. (33) are combined in one constant c3:
DTsh
1 0 1=3 2=3 vTeq rice dFeret @ cp;liq lliq w þ 1A Tw Tliq ¼ c3 ris f vw D2=3 Dhf r2=3 liq
c2 12BV ai with c3 ¼ c1 BA di;inner acr
ð34Þ
The ratio of the heat transfer coefficients in the expression for c3 is assumed constant here. The experiments with
Fig. 12 – Relation between variables at right-hand side of Eq. (34) and measured degrees of superheating; the numbers in the figure represent the experiment number as listed in Table 1.
different ice slurry velocities showed similar relative superheating values indicating that this assumption is reasonable. The experimental variables at the right-hand side of Eq. (34) are considered at the start of each experiment. This analysis is limited to the initial phase of the experiments, since the average ice crystal size was only determined prior to each experiment. It is expected that the average crystal size decreases in the course of an experiment, but this was not quantified. The results of this analysis for all the 10 melting experiments shown in Fig. 12 confirm proportionality between the variables and the degree of superheating stated in Eq. (34). The expression in Eq. (33) shows that the degree of superheating also depends on the tube diameter. According to the expression, the degree of superheating decreases with increasing tube diameter. Since the diameter of the heat exchanger tube was not varied in the experiments, this influence cannot be confirmed.
6.
Conclusions
The liquid temperature of ice slurry in melting heat exchangers can be significantly higher than the equilibrium temperature. This phenomenon is referred to as superheating and can lead to a serious reduction of heat exchanger capacities. The degree of superheating at the outlet of heat exchangers is proportional to the average ice crystal size and the heat flux. It is furthermore inversely proportional to the ice fraction and increases therefore as the ice fraction decreases. The negative growth rate of melting ice crystals is determined by both heat and mass transfer. The degree of superheating increases therefore with increasing solute concentration, especially at higher concentrations at which the heat transfer resistance plays a minor role. Finally, the degree of superheating is expected to be higher in heat exchangers with small hydraulic diameters.
920
international journal of refrigeration 31 (2008) 911–920
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