ENGINEERING, DESIGN, AND EQUIPMENT
Temperatur
s in
ids GEORGE J. KELLERl,
JOHN H. BALLARD2,
Wesfern Ufilizafion Research Branch, Pasadena, Calif. Universify of Southern California, Los Angeles, Calif.
OSLYN and Marsh ( 1 8 ) ieported cooling and heating curves for frozen sucrose solutions and fruit juices. Campbell, Proctor, and Sluder ( 3 ) reported values foi the thermal conductivity of frozen single-strength and 25' Brix concentrated orange juice. Riedel (32) derived a relationship among temperature, concentration, and thermal conductivity for fruit and vegetable juices, but the temperature range was all above 32" F., which is also above the freezing point of even single-strength juice where none of the complicating factors due t o ice formation are involved. Morgan ( 2 2 ) reported an attempt to apply the GurneyLurie ( 1 2 , 2 2 ) plots combined with charts of Hottel (17, 21), tables of Olson and Schultz ( 2 7 ) , and Newman's ( 2 3 ) method t o obtain values of thermal conductivity. H e studied concentrated orange juice of 42" and 58.9" Brix, but indicated his results were of questionable value. General methods (5-6,7 , 8, 19, 20, 24-28, 28-50, 54,39, 46) designed t o predict the freezing times of foods in various shapes all assume that freezing progresses by a frozen phase boundary moving through the freezing material and separating tlvo hypothetical zones-one of frozen and the other of unfrozen material. As this surface moves through the material, the latent heat is considered evolved a t this boundary and then conducted through the frozen layer to the surroundings. When the boundary has progressed through the object, any further reduction in temperature is treated as cooling a frozen material. This is not a reasonable concept of freezing or thawing a solution such as fruit juice. For all practical purposes fruit juices can be considered t o have the freezing properties of a typical twophase system: one for ice, and the other for solution. At a temperature below the initial freezing point, a state of equilibrium exists wherein a specific amount of ice will have formed and the remaining solution will have been altered in concentration by the removal of water as ice. At other temperatures equilibrium conditions exist with different amounts of ice in equilibrium with their respective amounts and concentrations of solution. Heat flowing into, or out of, an increment of frozen juice will not only 1 2
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raise or lower the temperaturP, hut mill also alter the amount of ice present and the amount as well as the concentration of the equilibrium solution. As the coniposition of the material changes, the ability t o conduct heat also changes. A unique condition exists throughout the object, depending upon the temperature a t any given position. It appears as a slushlike mixture composed of ice crystals and concentrate with a different amount of the initial water content as ice at each position. Schroeder and Cotton (58)found no evidence of a eutectic on freezing various concentrations of orange juice to -40" C. Heiss and Schachinger ( 1 6 ) predicted a eutectic for apple juice between -23" C. and -25' C. As cooling or warming occurs, therefore, latent heat is evolved or absorbed throughout the entire object as long as the temperature is changing belov its initial freezing point and is a function of the position. Moving boundary concepts, therefore, failed to provide a suitable approach to the problem, and although they may have served in certain instances, they could not be applied to solutions, Theory is evolved for fruit juices packaged in cylindrical shapes
One of the requirements for a solution is the representation of t h e freezing object as a relatively simple geometric shape. Fruit juices are usually packaged in cylindrical shapes, and this work deals only with this shape. Several simplifying assumptions can be made. The head space a t the top of a container constitutes a dead air space, or in cases where vacuum closure is employed, leaves an evacuated space, and is a poor conductor of heat. The c.linder is generally supported on end bv a floor or shelf, on top of another cylinder, or on some other poorly conducting material-eycept the original freezing equipment. The container can be considered a cylinder with thermally insulated ends, or as a cylinder of infinite length, and the assumption can be made that there is radial heat flow only. A differential equation for unsteady-state radial conduction of heat can be obtained and combined with a heat balance of an element of volume from a cylinder of frozen juice. Considering a differential element of frozen juice as shown in Figure 1, the following relationships apply: Heat entering,
Temperature distributions, during unsteady heat flow, through containers of frozen fruit and vegetable juices are predicted by a graphical method, representing the frozen object as a simple cylinder and assuming that there is radial heat flow only 188
Heat leaving.,
Heat of fusion, Heat stored,
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 2
ENGINEERING, DESIGN, AND EQUIPMENT A heat balance gives: Heat entering - heat leaving = heat stored.
- dqz
dqi
4
3
+ dq4
(5)
Figure 2, which represents the temperature distribution about a typical radial increment, n. The change of temperature in time A0 is:
Substituting, combining, and simplifying:
Aet = tn,e+l
- ( g ) T = a [ g +1r xbt& ] - B
P
k/c
x
1 - (t,+l,e 2
A,t
=
=
RF X AHFIc
p
(12)
The change of temperature in radius Ar is:
where Ly
- t,.e
(7) (8)
The negative sign on the left-hand side of the equation can be neglected for the purpose of this presentation.
- tn-1.0)
(13)
The rate a t which the change in temperature is changing in Ar is: bat = (tn+l,e
- tn,e) - ( L , s
- tn-l.e)
(14)
Substituting in Equation 11 gives:
p,+~.e t,-l,e
Ad =
1 4- 2n
- %.e
(&,+I.@
- tn-1,8
11 -
P X
(15)
A0
Reconsider Figure 2 with the logarithm of n plotted as abscissas, where n is the number of the increment, zero corresponds t o n = 1, and the center corresponds t o negative infinity. T h e zero point on this plot represents the first increment, Ar, from the center.
J dz
I
t
-ln(n-i)--+
Figure 1. Radial heat transfer in cylinder of frozen juice
In (n) In (n+ I)
and The direct integration of this equation is hopeless, as vary with temperature and radial position. However, a graphical integration developed by Grossman (10) for the solution of a similar equation for the radial temperature distribution in a cylindrical catalytic reactor suggested a graphical solution of Equation 6. Rewriting, Equation 6 in incremental form becomes :
Figure 2. Radii spacing for graphical calculation of temperature
From the geometry of Figure 2 Equation 15 can be satisfied graphically by constructing a line from A t o C and subtracting the quantity (3 X A0 t o give point E as follows:
FD
=
AG
CC
- In ( n ) - l n ( n - 1) - In ( n / n - 1) - In ( n + 1) - In (n - 1) n+ l n - 1
where Aet denotes a change in temperature in time A0, and A,t denotes a change of temperature in radial distance AT. The graphical solution is implemented by maintaining the following relationship between the radial distance and time of each incremental volume under consideration. A0 =
2ff
BE
=
BE =
FD
- FB - DE
(tn+l,8
- t,-l.s)
(i +
GC -
(i + &) -
in)
(tn,e
- tn-l.e)
(17)
B X A0
Equation 9 becomes:
(11) Adapting Grossman's (10) procedure, which is similar t o t h a t of Binder ( 1 ) and Schmidt (56-57),the radius of the cylinder is divided into n equal but finite increments, Ar-then, r = n X AT. Time is divided into 0 small increments, A0, related t o AT by Equation 10. The following relationships can be derived from
February 1956
- F B - DE
BE =
' [t " + i , ~+
2
- 2tn,0 + 2n ( t n + i . ~ - tn-1.8 1
tn-1.0
P
)I
X A0
(18)
(19)
Equation 19 is identical with Equation 15. The latent heat of fusion, AH=, is negative. When a heat gain is registered in a n elemental volume under consideration, some of the heat is absorbed by the melting ice and thereby
INDUSTRIAL AND ENGINEERING CHEMISTRY
189
ENGINEERING, DESIGN, AND EQUIPMENT reduces the net temperature increase of the material. When a heat loss occurs, some of the heat is evolved by the fusion of the ice and does not result in a teniperature lowering of the mateiial represented by the heat loss. Therefore, the quantity p X A0 results in a reduction in the temperature change, or a subtraction to give point E .
The solution of a problem where the surface conductance is involved, such as when the surface teniperature is not the same as the temperature of the surroundings, can be accommodated as foll0a.s: Consider the increment of volume shown in Figure 1 as bounded by the surface of the cylinder; then:
dql (leaving) dq, (entering)
=
- h A ( t , - t,,,)dO
=
[(E)
--k,
27ir X
dqa(stored) = -2~74 X d, X d z X
p
>
