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Freezing Fouling or Liquid Solidification
137
CHAPTER
9
Freezing Fouling or Liquid Solidification 9.1
INTRODUCTION
Where a flowing liquid is being cooled and the wall of the channel through which it flows is below the freezing point of the liquid, solidification of the liquid at the surface is likely to take place. The presence of this solid layer constitutes a resistance to heat removal from the flowing liquid. In many respects the phenomenon is similar to crystallisation fouling except that in principle, the diffusion or mass transfer of the dissolved solute towards the surface does not apply; the depositing molecules are already at the surface. The concept of freezing fouling has been applied to organic systems [Bott 1981, 1988] where there is a "spread" of molecular weight, but of essentially the same chemical family, e.g. the deposition paraffin wax during the cooling of waxy hydrocarbons or crude oils (see Chapter 8). The production of chilled water in the fine chemical manufacture and food processing industries may also give rise to freezing fouling where ice is formed on the cold surface. The problem may also exist in vapour systems during the recovery of solid products, e.g. the production of phthalic anhydride crystals in socalled "switch condensers". 9.2
CONCEPTS AND MATHEMATICAL ANALYSIS
The problem of freezing or liquid (water) solidification, has importance in aspects of climate, conservation in nature and soil mechanics. In general these represent static systems where the water is not flowing as in an industrial heat exchanger. Nevertheless, the study of static systems does give some insight into the approach that may be made for liquid systems subject to movement. 9.2.1
Static systems
Considerable experimental and analytical effort has been applied to the problems of phase change during heat removal from non-flowing systems [Riley et al 1974, Huang and Shih 1975, Stephen and Holzknecht 1976]. The basic problem is simple in concept (see Fig. 9.1), but difficult to apply to complex geometries. Considering a flat surface, the temperature of which (T~) is maintained below 0~ by a large heat sink in contact with stationary water, as illustrated in Fig. 9.1, it is possible to visualise the mechanism of ice formation.
138
Fouling o f tteat Exchangers
FIGURE 9. I. Ice formation at a surface below O~
Sensible heat and the latent heat of freezing are removed from the water at the liquid/solid interface. Under the prevailing static conditions heat will pass from the water to the cold sink by conduction. The resistance to this heat flow initially, will be a combination of the thermal resistances in the liquid and due to the cold solid. Immediately a layer of ice begins to form on the cold surface a further resistance is added to the other thermal resistances. As further heat is extracted from the water, the ice layer thickens representing an advancing boundary between the solid ice and the liquid water, i.e. a transient condition. The transient condition, coupled with complex geometries and different forms of ice structure dependent in turn on the rate of cooling, constitute severe problems of mathematical analysis. The ice/water interface will advance into the bulk water phase but as the thickness of the ice layer increases with the associated increase in thermal resistance, the rate of heat removal will fall. In turn this will represent an exponentially decreasing rate of growth of the ice layer. Referring to Fig. 9.1 if at an instant in time when the ice layer thickness is x the thickness of the ice layer increases by d~ in time dt the mass of ice formed per unit area is p,~, dr and the heat removed in forming this ice is h ~ p~, dr where h~, is the latent heat or enthalpy of fusion of ice and q = - T~-----~) (Ts ,~.~,
(9.1)
where TI is the interface between the water and the ice, i.e. normally 0~ . dt(Ty - T~) 2,,-h,~,p~,dr X
(9.2)
139
Freezing Fouling or Liquid Solidification
.'.dr=
(9.3)
Assuming that the specific heats of ice and water are small compared with the latent heat of fusion integrating yields: t=Kx 2
(9.4)
where K is a constant =
2(TI - T~).;1.,,
It will be seen that the constant K contains terms that are related to the thermal properties of the system namely
h,~, which may be given the collective AT2i,~
symbol ft. where AT represents the temperature difference (T~ - Tc). Riley et al [ 1974] used this group in a study of inward solidification of spheres and cylinders. The technique assumes constant thermal properties and that the value of fl is large. Values of fl used varied from 10 to 20. Stephen and Holzl~echt [ 1976] used a similar parameter that they called the phase conversion factor to solve solidification problems. The magnitude of the errors decreases as the phase conversion factor is increased although values much less than those quoted by Riley et al [ 1974] are claimed. In a very comprehensive paper Huang and Shih [1975] developed perturbation solutions for liquid solidification in a phase co-ordinate system. The technique adopted involved immobilising the moving solid/liquid boundary by the use of Landau's transformation [Bankoff 1964]. The time variable is replaced by the advancing interface and applying the perturbation technique. Referring again to Fig. 9.1 at an instant in time when the ice layer thickness is x and assuming that the temperature fall in the liquid from the bulk temperature of TB to the temperature at the ice/water interface T, (normally 0~ occurs over a distance y the rate of heat transfer is given by: q=
x
t-
y
(9.5)
~i~ ~wat~r where ~,~ and ,;L~ are the thermal conductivities of ice and water respectively.
140
Fouling of Heat Exchangers
Equation 9.5 assumes that the water is stationary, i.e. no convective heat transfer is present. An estimation of the heat removed is complex since it not only involves latent heat of fusion, but sensible heat effects that may not be insignificant where large systems are involved. A further complication arises where natural convection in the water at the water/ice interface occurs, i.e. modifying the simple conduction concept implied in Equation 9.5. The research of Burton and Bowen [1988] into the freezing of water in vertical tubes, clearly shows the complexities of freezing in static conditions. The work was initiated to provide a better understanding of ice plug growth during industrial pipeline maintenance, i.e. potential leakage is controlled by the development of a plug of frozen solid. Fig. 9.2 illustrate three phases of the freezing process identified by Burton and Bowen [ 1988].
FIGURE 9.2. Three phases of a freezing process
In the first phase freezing is most rapid near the bottom of the cold section. Transition to the second phase may be identified by a reduction in the rate of
Freezing Fouling or Liquid Solidification
141
freezing due to the mixing of the hot and cold components of the water, but at the same time ice continues to form in the upper parts of the freezing section, causing the restricted region to migrate upwards. The third phase occurs when a complete plug has formed when it extends in the axial direction. Changes in the pattern of natural convection are thought to account for these distinct changes. The convective flows will also be influenced by the density inversion of water at 4~ Although a study of liquid solidification in static systems is of interest, and gives some insight into the physical behaviour of freezing at a solid surface, the more complex situation that prevails in flowing systems is of more interest to the heat transfer engineer. 9.2.2. FLOWING SYSTEMS Fig. 9.3 illustrates the situation that exists at an instant of time when a flowing fluid (usually a liquid) subject to solidification, is cooled by another fluid flowing counter current on the other side of a metal wall.
FIGURE 9.3. Solidification in a flowing system
As a result of the temperature difference between the two fluids heat will flow from the liquid subject to freezing, across the metal wall and into the coolant. The rate of heat removal will of course be dependent upon the resistance to heat flow provided by the fluids themselves, the metal wall, the solid frozen layer, and any fouling resistance on the coolant side. At the beginning of the process, before any solid appears, the layers of warm fluid will flow in contact with the metal wall. In so doing they will lose sensible heat across the metal wall to the coolant. As heat is removed from these layers, the temperature of the liquid in contact with the cold metal surface will eventually fall to the freezing temperature of the liquid. Under these conditions solidification at the metal surface is possible. It is likely that some degree of subcooling will
142
Fouling of Heat Exchangers
occur before solidification begins, for similar reasons to those mentioned in Chapter 8, for crystallisation of a normal solubility salt. For turbulent conditions in the fluids with a reasonable finite temperature difference, the onset of solidification is likely to be rapid, since no diffusion of material to the cold metal surface is required before fouling of the surface commences. Immediately solid appears on the surface a further resistance to the transfer of heat is added to the system, thereby reducing the rate of heat transfer. The process of heat removal will continue till the temperature distribution between the "hot" fluid and the coolant is such that the outer layer of the frozen solid is just at the freezing point. Under these conditions, a steady state is reached when the solid layer remains at constant dimensions. The simple concept will be modified by the practical fouling effects that have already been noted in respect of fouling in general. Principally these effects include changes in velocity for a given flow rate as the deposition reduces the flow area, and the change in roughness presented to the flowing fluid by the deposit surface compared with the original substrate (metal) surface. The combination of these effects on the liquid being cooled, is to change the heat transfer coefficient between the bulk fluid on the solid layer. In turn this will affect the temperature distribution. The effects of the increased shear at the surface may also affect the equilibrium thickness of the solidified layer. The thickness of the layer is unlikely to be uniform since the temperature difference at equilibrium conditions will vary from the inlet of the freezing liquid to its outlet. The effect will be to change the position of the freezing temperature along the fluid path relative to the wall. Knowing the end temperatures of both streams it would be possible to calculate step changes along the wall and use an interactive procedure to make an estimate of the frozen layer along the wall. The procedure is complex and would most certainly require major simplifying assumptions. A review of some of the numerical and approximate methods that have been employed to gain mathematical insight into the problem of freezing fouling was made by Siegel and Savino [ 1966]. As a result they developed a one dimensional solidification model of a liquid flowing over a thin fiat plate. The plate is considered sufficiently thin so that the heat removal required to reduce the temperature of the plate is small compared with the heat flow through the wall. The heat flow through the plate is made up of three components; the latent heat of fusion of the solidifying material, the sensible heat removal from the deposited solid and the heat transferred by convection at the frozen layer/liquid interface. Siegel and Savino [1966] employed three analytical methods. Two of these techniques involve the integration of the transient heat conduction equation which governs the heat flow through the deposit, to provide a generalised form of the growth of the frozen layer with time. These analysts used techniques developed by Adams [1958] and Goodman [1958]. The work of Goodman also provided the basis for the third technique and involves a heat balance at the liquid/frozen layer interface.
Freezing Fouling or Liquid Solidification
143
For steady state conditions (i.e. constant frozen layer thicknesses) it may be assumed that the temeprature of the liquid/frozen layer interface is the freezing temperature TI and the only heat removal is the heat transferred across the resistances from the "hot" liquid to the coolant. Referring to Fig. 9.3 the wall temperatures of the hot and cold fluids are TH and Tc respectively. Under these conditions of steady state the heat flux, q = a M ( T n - TI )
(9.6)
where a H is the heat transfer coefficient at the "hot" liquid/frozen layer interface. The resistance to heat flow along the temperature gradient TI - Tr is: x,
x, + l
2,
2,
a,
where 2, and 2,, are the thermal conductivities of the frozen layer and metal respectively. a c is the heat transfer coefficient between the metal/coolant interface. therefore q = x , + xm + ~1 2,
/Z,
(9.7)
a,
Combining equations 9.6 and 9.7 yields an(Tn - T i ) -
TI - T~ x , ~ x,. + ~ 1 2,, ,~,, Ct,
(9.8)
Rearranging gives the value of the equilibrium frozen layer thickness
(1)
3,___~,(TI - T~) _ 2, xm +
(9.9)
Equation 9.9 indicates the effect of the variables on the thickness of the solid layer and confirms some intuitive observations. Increasing the liquid and coolant temperatures Tn and T~ and the heat transfer coefficient g~ will also reduce the frozen layer thickness. Opposite changes in the variables will have the converse effect. When no frozen layer exists x, = O so that equation 9.9 reduces to
144
Fouling o f Heat Exchangers
(9.10)
an(Tn _ TI) = A, Am + a,
TI-T~)
1
(9.11)
For freezing fouling to be prevented the bulk temperature of the hot fluid Tn must be(Ti - T~).
1
higher than the freezing temperature.
+
Ignoring the thermal resistance of the substrate (e.g. metals are good conductors of heat) Tn = TI + a~ ( TI _ T~)
(9.12)
a H
Two parameters can be introduced to facilitate the analysis of freezing fouling problems. Siegel and Sarino [1966] include the sub-cooling parameter S sometimes known as the Stefan number where
S -cp'(TI-T )
h,
(9.13)
cp, is the specific heat of the solid layer and the ratio of the heat transfer resistances R where
(9.14) ctc
~m
It may be noted that S is the inverse of the factor introduced by Riley et al [ 1974] for the analysis of static systems. The term cp, (TI - To) in the Stefan number represents the maximum internal energy it is possible to remove from unit mass of the solidified liquid in reducing its temperature from the freezing point down to the coolant temperature. If the
Freezing Fouling or Liquid Solidification
145
coolant temperature is reduced towards absolute zero, S will increase towards a maximum value. Siegel and Savino [1966] observe that for many materials, the maximum value of S is approximately 3. Xs
For thin layers of frozen solid, ~
tends to zero so that the value of the
parameter R will tend to zero. For steady state conditions a~ will be constant, x, is fixed and the physical properties ~, and Am will also be constant, so that the value of R is a measure of the solid layer thickness x,, under steady state conditions. The heat of fusion to be removed to produce a frozen layer thickness of x, is p,h,x,. If this freezing process takes place in time t the total heat transferred by convection from the hot fluid to the solid is t a n (Tn - Tr) per unit area. The ratio of these two quantities of heat may be used to define a dimensionless time t ~ so that t, =
p,h,x,
(9.15)
It is also possible to define an dimensionless frozen layer thickness as x~ = x x,
(9.16)
where x is the instantaneous frozen layer thickness. x ~will vary from zero when the metal surface is clean to unity when equilibrium has been established. A large value of dimensionless time t ~ suggests that the heat transfer by convection is large compared to the heat removed to create the foulant layer. For different sets of conditions within a system with a fixed value of the dimensionless thickness x ~, it is possible to plot a graph of dimensionless time t ~ against R the ratio of the heat transfer resistances for different values of S the Stefan number, as shown on Fig. 9.4, originally suggested by Siegel and Savino [1966].
146
oJ
E
Fouling of Heat Exchangers
Fixed dimensionless frozen layer thickness
,4,,=
el tit c o el
o..-.
QI
E
o ~
O
S=1.0 S=0.5 S=O
Thermal resistance parameter R FIGURE 9.4. Dimensionless time t I plotted against R for different values of S
The usefulness of figures similar to Fig. 9.4 for .particular systems is that they provide a means of estimating the frozen layer thickness at any time t. For a given set of conditions, where all the properties of the system are known. Temperatures T x and T c are specified, and the values of a n and a c may be calculated from the standard equations, the values of R, S and t ~may be estimated. Some inaccuracy in the calculation of a H may be introduced since the roughness of the deposit may not be readily assessed. Increased roughness will increase the value of all. Having plotted the graphs along the lines of Fig. 9.4 the relationship between x and t can be calculated from the equations for t ~and x ~. Using a jet of warm water directed at a plate maintained at a low temperature Savino et a l [1970] have compared the data obtained with those from the mathematical analysis. The comparison made on Fig. 9.5 shows excellent agreement between the experimental data and the theory. It is noteworthy that the equilibrium ice layer is reached after a relatively short time. The fall in wall temperature mirrors the increase in equilibrium ice layer thickness. The general shape of the accumulation of ice on the surface follows the exponential curve of deposit thickness with time often associated with fouling of heat exchanger surfaces.
Freezing Fouling or Liquid Solidification
147
4-0 \
I../ 0
-20 K..
x
3"0
t/t t/t
-I--
'-'. -40 r
&..I o..
E Q,I
.4-
"~ -60
Predicted ice thickness
z.0
O
Measured data
K.-
0
1.0
Wall temperature
/
r
0
0
4
8
12
16
Time from s t a r t minutes FIGURE 9.5 Experimental data compared to the mathematical model
REFERENCES Bankoff, S.G., 1964, Heat conduction or diffusion with change of phase. Advances in Chemical Engineering. Academic Press, New York. Bott, T.R., 1981, Fouling due to liquid solidification, in: Somerscales, E.F.C. and Knudsen, J.G. eds. Fouling of Heat Transfer Equipment. Hemisphere Publishing Corp. Washington. Bott, T.R., 1988, Crystallisation of organic materials, in" Melo, L.F., Bott, T.R. and Bemardo, C.A. eds. Fouling Science and Technology. Kluwer Academic Publishers, Dordrecht. Burton, M.J. and Bowen, R.J., 1988, Effect of convection on plug formation during cryogenic pipe freezing. 2nd UK Nat. Heat Transfer Conf. Vol. 1, 465 - 476, Glasgow. IMechE/IChemE. Goodman, T.R., 1958, The heat balance integral and its application to problems involving a change of phase. ASME, 80, 335 - 342. Huang, C.L. and Shih, S.Y., 1975, Perturbation solutions of planar diffusion controlled moving boundary problems. Int. J. Heat and Mass Transfer, 18, 689 - 695. Riley, D.S. et al, 1974, The inward solidification of spheres and cylinders. Int. J. Heat and Mass Transfer, 17, 1507 - 1516. Savino, J.M. et al, 1970, Experimental study of freezing and melting of flowing warm water at a stagnation point on a cold plate. 4th Int. Heat Transfer Conf. 1, 2 - 11.
148
Fouling of Heat Exchangers
Siegel, R. and Savino, J.M., 1966, An analysis of the transient solidification of a flowing warm liquid on a convectively cooled wall. 3rd Int. Heat Transfer Conf., 4, 141- 151. Stephen, K. and Holzkrecht, B, 1976, Die asymptotischen 16sungen ~ r vorgange des erstarrens. Int. J. Heat and Mass Transfer, 19, 597 - 602.
CHAPTER
9
Freezing Fouling or Liquid Solidification 9.1
INTRODUCTION
Where a flowing liquid is being cooled and the wall of the channel through which it flows is below the freezing point of the liquid, solidification of the liquid at the surface is likely to take place. The presence of this solid layer constitutes a resistance to heat removal from the flowing liquid. In many respects the phenomenon is similar to crystallisation fouling except that in principle, the diffusion or mass transfer of the dissolved solute towards the surface does not apply; the depositing molecules are already at the surface. The concept of freezing fouling has been applied to organic systems [Bott 1981, 1988] where there is a "spread" of molecular weight, but of essentially the same chemical family, e.g. the deposition paraffin wax during the cooling of waxy hydrocarbons or crude oils (see Chapter 8). The production of chilled water in the fine chemical manufacture and food processing industries may also give rise to freezing fouling where ice is formed on the cold surface. The problem may also exist in vapour systems during the recovery of solid products, e.g. the production of phthalic anhydride crystals in socalled "switch condensers". 9.2
CONCEPTS AND MATHEMATICAL ANALYSIS
The problem of freezing or liquid (water) solidification, has importance in aspects of climate, conservation in nature and soil mechanics. In general these represent static systems where the water is not flowing as in an industrial heat exchanger. Nevertheless, the study of static systems does give some insight into the approach that may be made for liquid systems subject to movement. 9.2.1
Static systems
Considerable experimental and analytical effort has been applied to the problems of phase change during heat removal from non-flowing systems [Riley et al 1974, Huang and Shih 1975, Stephen and Holzknecht 1976]. The basic problem is simple in concept (see Fig. 9.1), but difficult to apply to complex geometries. Considering a flat surface, the temperature of which (T~) is maintained below 0~ by a large heat sink in contact with stationary water, as illustrated in Fig. 9.1, it is possible to visualise the mechanism of ice formation.
138
Fouling o f tteat Exchangers
FIGURE 9. I. Ice formation at a surface below O~
Sensible heat and the latent heat of freezing are removed from the water at the liquid/solid interface. Under the prevailing static conditions heat will pass from the water to the cold sink by conduction. The resistance to this heat flow initially, will be a combination of the thermal resistances in the liquid and due to the cold solid. Immediately a layer of ice begins to form on the cold surface a further resistance is added to the other thermal resistances. As further heat is extracted from the water, the ice layer thickens representing an advancing boundary between the solid ice and the liquid water, i.e. a transient condition. The transient condition, coupled with complex geometries and different forms of ice structure dependent in turn on the rate of cooling, constitute severe problems of mathematical analysis. The ice/water interface will advance into the bulk water phase but as the thickness of the ice layer increases with the associated increase in thermal resistance, the rate of heat removal will fall. In turn this will represent an exponentially decreasing rate of growth of the ice layer. Referring to Fig. 9.1 if at an instant in time when the ice layer thickness is x the thickness of the ice layer increases by d~ in time dt the mass of ice formed per unit area is p,~, dr and the heat removed in forming this ice is h ~ p~, dr where h~, is the latent heat or enthalpy of fusion of ice and q = - T~-----~) (Ts ,~.~,
(9.1)
where TI is the interface between the water and the ice, i.e. normally 0~ . dt(Ty - T~) 2,,-h,~,p~,dr X
(9.2)
139
Freezing Fouling or Liquid Solidification
.'.dr=
(9.3)
Assuming that the specific heats of ice and water are small compared with the latent heat of fusion integrating yields: t=Kx 2
(9.4)
where K is a constant =
2(TI - T~).;1.,,
It will be seen that the constant K contains terms that are related to the thermal properties of the system namely
h,~, which may be given the collective AT2i,~
symbol ft. where AT represents the temperature difference (T~ - Tc). Riley et al [ 1974] used this group in a study of inward solidification of spheres and cylinders. The technique assumes constant thermal properties and that the value of fl is large. Values of fl used varied from 10 to 20. Stephen and Holzl~echt [ 1976] used a similar parameter that they called the phase conversion factor to solve solidification problems. The magnitude of the errors decreases as the phase conversion factor is increased although values much less than those quoted by Riley et al [ 1974] are claimed. In a very comprehensive paper Huang and Shih [1975] developed perturbation solutions for liquid solidification in a phase co-ordinate system. The technique adopted involved immobilising the moving solid/liquid boundary by the use of Landau's transformation [Bankoff 1964]. The time variable is replaced by the advancing interface and applying the perturbation technique. Referring again to Fig. 9.1 at an instant in time when the ice layer thickness is x and assuming that the temperature fall in the liquid from the bulk temperature of TB to the temperature at the ice/water interface T, (normally 0~ occurs over a distance y the rate of heat transfer is given by: q=
x
t-
y
(9.5)
~i~ ~wat~r where ~,~ and ,;L~ are the thermal conductivities of ice and water respectively.
140
Fouling of Heat Exchangers
Equation 9.5 assumes that the water is stationary, i.e. no convective heat transfer is present. An estimation of the heat removed is complex since it not only involves latent heat of fusion, but sensible heat effects that may not be insignificant where large systems are involved. A further complication arises where natural convection in the water at the water/ice interface occurs, i.e. modifying the simple conduction concept implied in Equation 9.5. The research of Burton and Bowen [1988] into the freezing of water in vertical tubes, clearly shows the complexities of freezing in static conditions. The work was initiated to provide a better understanding of ice plug growth during industrial pipeline maintenance, i.e. potential leakage is controlled by the development of a plug of frozen solid. Fig. 9.2 illustrate three phases of the freezing process identified by Burton and Bowen [ 1988].
FIGURE 9.2. Three phases of a freezing process
In the first phase freezing is most rapid near the bottom of the cold section. Transition to the second phase may be identified by a reduction in the rate of
Freezing Fouling or Liquid Solidification
141
freezing due to the mixing of the hot and cold components of the water, but at the same time ice continues to form in the upper parts of the freezing section, causing the restricted region to migrate upwards. The third phase occurs when a complete plug has formed when it extends in the axial direction. Changes in the pattern of natural convection are thought to account for these distinct changes. The convective flows will also be influenced by the density inversion of water at 4~ Although a study of liquid solidification in static systems is of interest, and gives some insight into the physical behaviour of freezing at a solid surface, the more complex situation that prevails in flowing systems is of more interest to the heat transfer engineer. 9.2.2. FLOWING SYSTEMS Fig. 9.3 illustrates the situation that exists at an instant of time when a flowing fluid (usually a liquid) subject to solidification, is cooled by another fluid flowing counter current on the other side of a metal wall.
FIGURE 9.3. Solidification in a flowing system
As a result of the temperature difference between the two fluids heat will flow from the liquid subject to freezing, across the metal wall and into the coolant. The rate of heat removal will of course be dependent upon the resistance to heat flow provided by the fluids themselves, the metal wall, the solid frozen layer, and any fouling resistance on the coolant side. At the beginning of the process, before any solid appears, the layers of warm fluid will flow in contact with the metal wall. In so doing they will lose sensible heat across the metal wall to the coolant. As heat is removed from these layers, the temperature of the liquid in contact with the cold metal surface will eventually fall to the freezing temperature of the liquid. Under these conditions solidification at the metal surface is possible. It is likely that some degree of subcooling will
142
Fouling of Heat Exchangers
occur before solidification begins, for similar reasons to those mentioned in Chapter 8, for crystallisation of a normal solubility salt. For turbulent conditions in the fluids with a reasonable finite temperature difference, the onset of solidification is likely to be rapid, since no diffusion of material to the cold metal surface is required before fouling of the surface commences. Immediately solid appears on the surface a further resistance to the transfer of heat is added to the system, thereby reducing the rate of heat transfer. The process of heat removal will continue till the temperature distribution between the "hot" fluid and the coolant is such that the outer layer of the frozen solid is just at the freezing point. Under these conditions, a steady state is reached when the solid layer remains at constant dimensions. The simple concept will be modified by the practical fouling effects that have already been noted in respect of fouling in general. Principally these effects include changes in velocity for a given flow rate as the deposition reduces the flow area, and the change in roughness presented to the flowing fluid by the deposit surface compared with the original substrate (metal) surface. The combination of these effects on the liquid being cooled, is to change the heat transfer coefficient between the bulk fluid on the solid layer. In turn this will affect the temperature distribution. The effects of the increased shear at the surface may also affect the equilibrium thickness of the solidified layer. The thickness of the layer is unlikely to be uniform since the temperature difference at equilibrium conditions will vary from the inlet of the freezing liquid to its outlet. The effect will be to change the position of the freezing temperature along the fluid path relative to the wall. Knowing the end temperatures of both streams it would be possible to calculate step changes along the wall and use an interactive procedure to make an estimate of the frozen layer along the wall. The procedure is complex and would most certainly require major simplifying assumptions. A review of some of the numerical and approximate methods that have been employed to gain mathematical insight into the problem of freezing fouling was made by Siegel and Savino [ 1966]. As a result they developed a one dimensional solidification model of a liquid flowing over a thin fiat plate. The plate is considered sufficiently thin so that the heat removal required to reduce the temperature of the plate is small compared with the heat flow through the wall. The heat flow through the plate is made up of three components; the latent heat of fusion of the solidifying material, the sensible heat removal from the deposited solid and the heat transferred by convection at the frozen layer/liquid interface. Siegel and Savino [1966] employed three analytical methods. Two of these techniques involve the integration of the transient heat conduction equation which governs the heat flow through the deposit, to provide a generalised form of the growth of the frozen layer with time. These analysts used techniques developed by Adams [1958] and Goodman [1958]. The work of Goodman also provided the basis for the third technique and involves a heat balance at the liquid/frozen layer interface.
Freezing Fouling or Liquid Solidification
143
For steady state conditions (i.e. constant frozen layer thicknesses) it may be assumed that the temeprature of the liquid/frozen layer interface is the freezing temperature TI and the only heat removal is the heat transferred across the resistances from the "hot" liquid to the coolant. Referring to Fig. 9.3 the wall temperatures of the hot and cold fluids are TH and Tc respectively. Under these conditions of steady state the heat flux, q = a M ( T n - TI )
(9.6)
where a H is the heat transfer coefficient at the "hot" liquid/frozen layer interface. The resistance to heat flow along the temperature gradient TI - Tr is: x,
x, + l
2,
2,
a,
where 2, and 2,, are the thermal conductivities of the frozen layer and metal respectively. a c is the heat transfer coefficient between the metal/coolant interface. therefore q = x , + xm + ~1 2,
/Z,
(9.7)
a,
Combining equations 9.6 and 9.7 yields an(Tn - T i ) -
TI - T~ x , ~ x,. + ~ 1 2,, ,~,, Ct,
(9.8)
Rearranging gives the value of the equilibrium frozen layer thickness
(1)
3,___~,(TI - T~) _ 2, xm +
(9.9)
Equation 9.9 indicates the effect of the variables on the thickness of the solid layer and confirms some intuitive observations. Increasing the liquid and coolant temperatures Tn and T~ and the heat transfer coefficient g~ will also reduce the frozen layer thickness. Opposite changes in the variables will have the converse effect. When no frozen layer exists x, = O so that equation 9.9 reduces to
144
Fouling o f Heat Exchangers
(9.10)
an(Tn _ TI) = A, Am + a,
TI-T~)
1
(9.11)
For freezing fouling to be prevented the bulk temperature of the hot fluid Tn must be(Ti - T~).
1
higher than the freezing temperature.
+
Ignoring the thermal resistance of the substrate (e.g. metals are good conductors of heat) Tn = TI + a~ ( TI _ T~)
(9.12)
a H
Two parameters can be introduced to facilitate the analysis of freezing fouling problems. Siegel and Sarino [1966] include the sub-cooling parameter S sometimes known as the Stefan number where
S -cp'(TI-T )
h,
(9.13)
cp, is the specific heat of the solid layer and the ratio of the heat transfer resistances R where
(9.14) ctc
~m
It may be noted that S is the inverse of the factor introduced by Riley et al [ 1974] for the analysis of static systems. The term cp, (TI - To) in the Stefan number represents the maximum internal energy it is possible to remove from unit mass of the solidified liquid in reducing its temperature from the freezing point down to the coolant temperature. If the
Freezing Fouling or Liquid Solidification
145
coolant temperature is reduced towards absolute zero, S will increase towards a maximum value. Siegel and Savino [1966] observe that for many materials, the maximum value of S is approximately 3. Xs
For thin layers of frozen solid, ~
tends to zero so that the value of the
parameter R will tend to zero. For steady state conditions a~ will be constant, x, is fixed and the physical properties ~, and Am will also be constant, so that the value of R is a measure of the solid layer thickness x,, under steady state conditions. The heat of fusion to be removed to produce a frozen layer thickness of x, is p,h,x,. If this freezing process takes place in time t the total heat transferred by convection from the hot fluid to the solid is t a n (Tn - Tr) per unit area. The ratio of these two quantities of heat may be used to define a dimensionless time t ~ so that t, =
p,h,x,
(9.15)
It is also possible to define an dimensionless frozen layer thickness as x~ = x x,
(9.16)
where x is the instantaneous frozen layer thickness. x ~will vary from zero when the metal surface is clean to unity when equilibrium has been established. A large value of dimensionless time t ~ suggests that the heat transfer by convection is large compared to the heat removed to create the foulant layer. For different sets of conditions within a system with a fixed value of the dimensionless thickness x ~, it is possible to plot a graph of dimensionless time t ~ against R the ratio of the heat transfer resistances for different values of S the Stefan number, as shown on Fig. 9.4, originally suggested by Siegel and Savino [1966].
146
oJ
E
Fouling of Heat Exchangers
Fixed dimensionless frozen layer thickness
,4,,=
el tit c o el
o..-.
QI
E
o ~
O
S=1.0 S=0.5 S=O
Thermal resistance parameter R FIGURE 9.4. Dimensionless time t I plotted against R for different values of S
The usefulness of figures similar to Fig. 9.4 for .particular systems is that they provide a means of estimating the frozen layer thickness at any time t. For a given set of conditions, where all the properties of the system are known. Temperatures T x and T c are specified, and the values of a n and a c may be calculated from the standard equations, the values of R, S and t ~may be estimated. Some inaccuracy in the calculation of a H may be introduced since the roughness of the deposit may not be readily assessed. Increased roughness will increase the value of all. Having plotted the graphs along the lines of Fig. 9.4 the relationship between x and t can be calculated from the equations for t ~and x ~. Using a jet of warm water directed at a plate maintained at a low temperature Savino et a l [1970] have compared the data obtained with those from the mathematical analysis. The comparison made on Fig. 9.5 shows excellent agreement between the experimental data and the theory. It is noteworthy that the equilibrium ice layer is reached after a relatively short time. The fall in wall temperature mirrors the increase in equilibrium ice layer thickness. The general shape of the accumulation of ice on the surface follows the exponential curve of deposit thickness with time often associated with fouling of heat exchanger surfaces.
Freezing Fouling or Liquid Solidification
147
4-0 \
I../ 0
-20 K..
x
3"0
t/t t/t
-I--
'-'. -40 r
&..I o..
E Q,I
.4-
"~ -60
Predicted ice thickness
z.0
O
Measured data
K.-
0
1.0
Wall temperature
/
r
0
0
4
8
12
16
Time from s t a r t minutes FIGURE 9.5 Experimental data compared to the mathematical model
REFERENCES Bankoff, S.G., 1964, Heat conduction or diffusion with change of phase. Advances in Chemical Engineering. Academic Press, New York. Bott, T.R., 1981, Fouling due to liquid solidification, in: Somerscales, E.F.C. and Knudsen, J.G. eds. Fouling of Heat Transfer Equipment. Hemisphere Publishing Corp. Washington. Bott, T.R., 1988, Crystallisation of organic materials, in" Melo, L.F., Bott, T.R. and Bemardo, C.A. eds. Fouling Science and Technology. Kluwer Academic Publishers, Dordrecht. Burton, M.J. and Bowen, R.J., 1988, Effect of convection on plug formation during cryogenic pipe freezing. 2nd UK Nat. Heat Transfer Conf. Vol. 1, 465 - 476, Glasgow. IMechE/IChemE. Goodman, T.R., 1958, The heat balance integral and its application to problems involving a change of phase. ASME, 80, 335 - 342. Huang, C.L. and Shih, S.Y., 1975, Perturbation solutions of planar diffusion controlled moving boundary problems. Int. J. Heat and Mass Transfer, 18, 689 - 695. Riley, D.S. et al, 1974, The inward solidification of spheres and cylinders. Int. J. Heat and Mass Transfer, 17, 1507 - 1516. Savino, J.M. et al, 1970, Experimental study of freezing and melting of flowing warm water at a stagnation point on a cold plate. 4th Int. Heat Transfer Conf. 1, 2 - 11.
148
Fouling of Heat Exchangers
Siegel, R. and Savino, J.M., 1966, An analysis of the transient solidification of a flowing warm liquid on a convectively cooled wall. 3rd Int. Heat Transfer Conf., 4, 141- 151. Stephen, K. and Holzkrecht, B, 1976, Die asymptotischen 16sungen ~ r vorgange des erstarrens. Int. J. Heat and Mass Transfer, 19, 597 - 602.