Heat transfer measurements were conducted during freezing of quiescent water in a horizontal cylinder. A horizontal cylinder with inner diameter of 61.1 mm is cooled by air in a constant low temperature room and time variations of the radial distribution of fluid temperature were observed. Experimental results for the velocity of the phase change interface, the time taken for complete freezing and apparent freezing heat transfer coefficient were compared with the simple theoretical model based on the quasisteady assumption

Feng, Yibu

Inaba, Hideo

Nozu, Shigeru

English

Okayama University Scientific Achievement Repository

Memoirs of the Faculty of Engineering, Okayama University,Vol.26, No.2, pp.17 -24, March] 992Freezing of Quiescent Water in a Horizontal Cylinder*. ** . **Yibu FENG , Hldeo INABA and Shlgeru NOZU(Received January 8 , ]992)SYNOPSISHeat transfer measurements were conducted during freezing ofquiescent water in a horizontal cylinder. A horizontal cylinderwith inner diameter of 61.1 mm is cooled by air in a constant lowtemperature room and time variations of the radial distribution offluid temperature were observed. Experimental results for thevelocity of the phase change interface, the time taken for completefreezing and apparent freezing heat transfer coefficient werecompared with the simple theoretical model based on the quasi-steady assumption.INTRODUCTIONFreezing of quiescent water in an enclosure is an important problem in relationto the cold heat storage(1) and food engineering, etc. A comprehensive review ofthe freezing model is given in Hattori(2). The phase change process is essentiallyunsteady. A model based on the quasi-steady assumption, however, usually gives agood prediction if the latent heat of freezing is larger than the sensible one. Thepresent study was undertaken to confirm the model based on the quasi-steadyassumption during freezing of quiescent water in a horizontal cylinder.* Department of Food Engineering, Heilongjiang Commercial College** Department of Mechanical Engineering1718 Yibu FENG, Hideo lNABA and Shigeru NOZUANALYSIS BASED ON QUASI-STEADY ASSUMPTIONConsider the growth rate of the annular ice layer in a horizontal cylinder withinner radius of Ri and the outer one of Ro ' The heat liberated at the phase changeinterface flows outward through the ice layer and tube wall to the ambience. If thefluid is kept at its freezing temperature t f during the phase change and also thewall resistance can be neglected in comparison with the other ones, an energybalance can be written aswhere y is the radial distance measured from the inner tube wall to the phase changeinterface, r = Ri - Y is the radius of the phase change interface, T is the time, toois the ambient temperature, ao is the heat transfer coefficient at the outer surfaceof the cylinder. Physical properties L, Pi and Ai are the latent heat of freezing,ice density and thermal conductivity of ice, respectively. The above equation canbe reduced todyr--dT(1)If the relation £n(Ri/r)/Ai « 1/(Roao ) holds (e.g., the cylinder is located in alow temperature room without air flow effects), equation (1) gives the followingequivalent resultsordyr--dTdy(Ri - y)-dTt f - too----RoaoPiLt f - too----RoaoPiL(2)(3)Solution to equation (3) subject to the initial condition, y o at T 0, givesy = Ri - VR/ - 2(tf - too)RoaoT/PiLTime taken for complete freezing T can be obtained by substituting yequation (4) as(4)Freezing of Quiescent Water in a Horizontat Cytimer 19T = (5)Apparent freezing heat transfer coefficient a i is defined on the basis of the phasechange interface basis as(6)Equation (6) indicates that the a i value depends only on r value and takes a minimumat r min given byEXPERIMENTrmin (7)Experimental Apparatus and Procedure Schematic view of the test cylinder is shownin Figure 1. The cylinder, filled with water, is placed horizontally in a constantlow temperature room and is cooled by air. The cylinder is made of copper withdimensions as follows: outer diameter, 63.1 mm; wall thickness, 1.0 mrn; tube length,500 mrn. Both ends of the cylinder are covered with rubber sheets with 1 mrn inthickness to ensure that abrupt pressure rise due to freezing is avoided.Eleven enamel-coated type-K thermocouples are used for measuring the radialdistributions of the fluid temperature. A thin acrylic rod with these thermocouplesis fixed at the mid cross-section of the cylinder. Thermocouple installation isshown in Table 1. Tube wall temperature is observed by the same type one solderedto the outer surface of the cylinder. Ambient air temperature variation around the-,...-....'"II \-____ C'?----CD \'S'"- - r-250500Figure 1 Side view of test cylinder20 Yibu FENG. Hideo INABA and Shigeru NOZUTable 1 Installation of thermocouple for measuring radial temperature distributionNo of thermocouple 2 3 4 5 6 7 8 9 10 11Distance measuredfrom inner tube 3.05 5.55 8.05 10.55 13.05 15.55 18.05 20.55 23.05 25.55 30.55wall (mm)cylinder are also measured. The outer surface of the cylinder is thermallyinsulated by fiberglass insulating material.Preliminary experiments, conducted using transparent end-sheets for the rubbersheets, revealed that one-dimensional radial heat flow is considered to bepredominant since little axial variation of the phase change interface was observedduring the freezing process. It is also confirmed that the measurements with highaccuracy were carried out since an agreement of within 12.3% was found between theobserved heat transfer rate and the calculated total amount of the latent heat offreezing (481.4 kJ).Results and Discussion Tables 2 and 3 summarize the results obtained at too = _18°Cand -28°C, respectively. In Tables 2 and 3, values of qL' ai and £ are obtainedrespectively fromand27TRoa o(tw - too)qL/{27Tr(t f - too)}£ = 1 - (r/R. ) 2l(8)(9)(10)where qL' a i and £ are the heat transfer rate at the tube surface per tube length,the apparent freezing heat transfer coefficient and the ratio of the cross-sectionalarea of the ice layer to the tube cross-sectional value, respectively. The resultsare compared with the prediction method described in the previous chapter.Table 2 Measured results obtained at ambient temperature of too _18°CT hr 2.5 4.38 6.84 8.25 9.05 10.12 10.55 10.75 11.39 11.75 12.19Y mm 3.05 5.55 8.05 10.55 13.05 15.55 18.05 19.31 20.55 23.05 25.55 30.55t w °C -0.4 -0.8 -1.3 -1.6 -1.8 -2.0 -2.5 -2.5 -2.9 -3.8 -4.1(10 W/(m2 K) 5.94 5.94 5.83 5.81 5.79 5.77 5.72 5.72 5.69 5.62 5.57ql W/m 20.69 20.22 19.27 18.86 18.57 18.27 17.55 17.55 17.01 16.02 15.32(Ii W/(m 2 K) 299.4 160.9 118.0 107.2 109.5 116.3 99.4 111.7 124.5 141.6£ % 19.0 33.0 57.1 67.2 75.9 83.3 86.5 89.3 94.0 97.3 100.0Freezing of Quiescent Water in a Horizantal Cylinder 21Table 3 Measured results obtained at ambient temperature of too = -28·CT hr 1.6 2.82 3.36. 3.98 4.66 5.52 6.26 6.5 6.68 6.98 7.20 7.58y mm 3.05 5.55 8.05 10.55 13.05 15.55 18.05 19.31 20.55 23.05 25.55 30.55t" ·C -0.5 -1.0 -1.1 -1.2 -1.7 -2.2 -2.8 -3.3 -3.5 -3.7 -3.7 -4.7ao W/(m2 K) 6.65 6.62 6.62 6.61 6.58 6.55 6.51 6.51 6.46 6.45 6.45 6.38q.l' W/m 36.20 35.38 35.25 35.06 34.25 33.45 32.47 31.58 31.33 31.02 31.02 29.42a i W/(m 2K) 419.0 225.2 226.7 232.5 183.2 161.3 147.7 135.5 142.5 177.9 266.9E % 19.0 33.0 45.8 57.1 67.2 75.9 83.3 86.5 89.3 94.0 97.3 100.0Figure 2 shows the effects of the ambient temperature too on the variation ofthe phase change interface with time. In Fig.2, lines I and 2 show the comparisonat too = -28·C and lines 3 and 4 at too = -18·C. It is seen from Fig. 2 that theobserved velocity of the phase change interface (lines 2 and 4), dy/dT, increasesmonotonically with increasing T, and the dy/dT value at too = -28·C is larger thanthat at too = -18·C. In both cases, the observed dy/dT value is somewhat smallerthan the prediction of equation (4) so that the observed time taken for completefreezing (T value at y = 30.55 mm) is somewhat longer than the prediction ofequation (5). A possible source of the discrepancies between the prediction and themeasurement includes that the thermal conductance of the ice layer is neglected inthe present analysis (average ratio of the conductance of the ice layer to the wall-to-ambient value is estimated to be less than 5%).Figure 3 shows the variation of the apparent freezing heat transfer coefficientUi with radius of phase change interface r, where line I shows the prediction ofequation (6), and lines 2 and 3 the observed value at too = -28 and -18·C,3530425E 2UE1510~ 51 2 3 4 5 0 7 8 9 101112 13T: hrFigure 2 Variation of phase change interface with time22 Yibu FENG. Hideo INABA and Sbigeru NOZU32mmr4-10420400381136034032030028026024022020018016014012010080 l...-...l---i_.J-.--I.._l--....L-~" 5 10 15 20 25' 30 35(11)heat transfer coefficient into equation (11).predicted by the equation.Neglecting the effect of the walltube wall temperatures t w' where the latterwas obtained by substituting a conventionalcorrelation equation of natural convectionresistance, a i can be related to ao asin trend, but show a dependence on too notFigure 4 compares the observed and predictedsets show a good agreement with equation (6)In Fig.4, lines 1 and 2 show the comparisonsharp decrease with decreasing of r, and thenit takes a minimum and shows a rapid upturnrespectively. The ai value first show awith further decreasing of r. The two dataat too = _28°C and lines 3 and 4 at too = -18t. The observed t w value at too = _28°C (line1) decreases more rapidly than that for too =Figure 3 Relationship betweenapparent freezing heat transfercoefficient and radius of phasechange interfacer hr0 1 2 3 4 U 6 7 8 9 101112130-1S-'-2-3~+J -4-5Figure 4 Variation of tube wall temperature with timeFreezing of Quiescent Water in a Horizontal Cylinder 2340E"30~320CJ1 4100 1 2 3 4 5 0 7 8 9 10 11 12 13T hrFigure 5 Variation of heat transfer rate per tube length with time10080~ GO4020_/IIlr-- 4-1 2 3 4 5 6 7 8 9 10 11 12 13hrFigure 6 Variation of cold heat stored with timeFigure 5 shows the variation of qf with T, where lines 1 and 3 are thepredictions obtained from equation (S) and lines 2 and 4 are the observed results.The observed value decreases moderately with increasing T, and consequently thendecreases rapidly near the end of the phase change.Figure 6 shows the ratio of the cross-sectional area of the ice layer to thetube value plotted on the coordinates of £ and T. Lines 1 and 3 show the predictionof equation (10) for too = -2S and -lS"C, respectively. The observed results take24 Yibu FENG, Hideo INABA and Shigeru NOZUlower E value than the prediction, with the difference increasing with T.CONCLUSIONSHeat transfer measurements have been conducted during freezing of quiescentwater in a horizontal cylinder cooled by natural convection of air. The resultshave been compared with a simple analytical model for predicting the freezingprocess. Conclusions obtained in the present study can be summarized as:1. The velocity of the phase change interface increases as the freezingproceeds. The radial distan~e y measured from the tube wall to the phase changeinterface can be calculated using equation (4). Also the time taken for completefreezing T can be obtained from equation (5).2. The apparent freezing heat transfer coefficient ui depends only on the yvalue and takes a minimum value at the position given by equation (7). The tubewall temperature can be calculated using equations (6) and (11).3. The ratio of the cross-sectional area of the ice layer to the tube value, E,increases lineally with time. The E value can be obtained from equations (4) and(10).4. The quasi-steady model can be used to make an approximate estimate for thefreezing process as that studied in the present paper.REFERENCES[1] M.Yanadori and T.Nasuda, Study on a Heat Storage Container with Phase ChangeMaterial (Part 3. Heat Transfer in a Cubic Heat Storage Container),Refrigeration, Vol.57, No.658(1982),799.[2] M.Hattori, Heat Transfer with Freezing and/or Melting, Refrigeration, Vol.62,No.714(1987),362.

Freezing of Quiescent Water in a Horizontal Cylinder

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