the objective of this work is to determine the best temperature regimes of ice production on a surface of the pivotal crystallizer. A computer program was developed to simulate the appropriate heat transfer equations that calculates the ice mass flow and freezing time by changing the evaporating temperature of refrigerant. This paper attempts to analyze the effect of evaporating temperature of refrigerant which flow through the pivotal crystallizer (evaporator) on the rate of ice mass flow and freezing time. The results obtained show that the lower evaporating temperature, the lower loss in ice mass flow during the process of ice growth on surface of the pivotal evaporator. Certainly the results of this study showed that the growth of ice block from ( 0.06M to 0.08M ) on surface of pivotal evaporator with evaporating temperature (-20c) causes decreasing the rate of ice freezing about 65%.  At the same  time  Using evaporating temperature (-5c) for the same growth of ice block will causes decreasing the rate of ice freezing about 72%.
 

Introduction

The first concern during the process of ice making on a surface of the pivotal evaporator is to maintain ice mass flow rate as high as possible. The rate of ice mass flow is considerably depends on the evaporating temperature of refrigerant. Therefore a careful study of evaporating temperature effect on ice mass flow rate will raise the efficiency of ice making process. It is known that the lower evaporating temperature, the greater ice mass flow rate. But a growth of ice block on a surface of the pivotal evaporator causes decreasing of ice mass flow rate as a result of ice layer resistance. But the decreasing rate of ice mass flow is not the same at different evaporating temperature. In order to find out the decreasing rate of ice mass flow at different evaporating temperature and at given ice diameter of ice block during the process of ice growth on a surface of pivotal evaporator, the decreasing rate must be determined for the given conditions. This can be achieved either by experiment or by computer based simulation. The later saves time and money. That is why I use computer simulation to study the effect of evaporating temperature on a loss of ice mass flow rate to give recommendation about the best evaporating temperature.

Heat transfer consideration

   The heat transfer from water to the interface during the process of ice growth is occurred by free convection. So the correlation suggested for natural convection heat transfer are given as follows:
For vertical surface (cylindrical or flat) [2] is
For laminar flow

For turbulent flow

    Where (Ra) is Rayleigh number defined as:

( pr_w ) is Prandtl number at water temperature as
( Pr_if ) is Prandtl number at interface temperature as
is Volumetric thermal expansion coefficient of water
( γ ) is Kinematic viscosity of water at water temperature
( a ) is Thermal diffusivity of water at water temperature
( H ) is Characteristic length of ice block
( Nu ) is Nusselt number expressed as:

   Equation (3) is used to determine Ra. Once that is found then Nusselt number is determined according to equations (1) and (2). Then heat transfer coefficient (hc), due to natural convection between water and interface is found using Eq. (4), as follows:

   Where ( K_w ) is Conductivity of water defined as:

   Where (tw) is water temperature
The heat balance equation for a process of ice freezing on a surface of the pivotal evaporator is written for average rate of heat flow per metre length, as follows:

   Where (Q_cd.a) Is the average rate of heat flow per metre length from the interface to the pivotal evaporator
by conduction expressed as:

   Where
( t_if ): interface temperature
( t_e ): evaporating temperature
( K_i ): conductivity of ice block
( D_I ): ice diameter
( D_e ): evaporator diameter
(Q_cv.a ) is the average rate of heat flow per metre length from the water to the interface
 of two phase by convection expressed as:

(Q_fr.a) is the average rate of heat flow of freezing per metre length defined as:

   Where
The value of (306(10^6)) is the heat of freezing (J/m³).
The freezing time, hour, (τ) determined as follows:
Equation (8) is used to determine (Q_cd.a) and equation (9) is used to determine (Q_cv.a). Once that are found then (τ) is determined using the heat balance equation (7), where equation (10) which is used to determine (Qfr.a) is taken into account.
   Thus

Since the freezing time has been found by equation (11). then the average ice mass flow rate is determined as follows:

The condition of finishing the process of ice growth is determined as follows:

   Where
(Q_cd) Is the rate of heat flow per metre length by conduction defined as: 

(Q_cv) Is the rate of heat flow per metre length by convection expressed as:

Once (Qcd) and (Qcv) are determined, then the maximum diameter of ice block is obtained by using Eq. (26) as follows:

   To determine the percentage of decreasing of ice mass flow rate, the following equation is used

   Where
Ml(i) is the initial  ice mass flow rate at (i) diameter of ice block.
Where i=0.06M
Ml(i+1) is the final ice mass flow rate at (i+0.01) diameter of ice block.

Simulation results

   The result of this study are based on (5c) water temperature, (0.05M) evaporator  diameter and (1M) evaporator length. Using the developed model and giving different evaporating temperature and ice block diameter, the average rate of ice mass flow (Mi), the freezing time,), and the maximum diameter of ice block are calculated and showing in table (1).

The ice mass flow rate in (kg/s) is illustrated as a function of ice block diameter and evaporating temperature in figure (1).
The maximum diameter of ice block is illustrated as a function of evaporating temperature in figure (2).
The average heat flow per metre length from the interface    to   pivotal evaporator   by conduction (Qcd.a) is illustrated as a function of  evaporating temperature and ice block diameter.

te (C)	Di (M)	Mi (kg/s)	Τ (hour)	Di max (m)
-5	0.06	0.00099	0.11	0.09
-5	0.09	0.00019	3.12
-10	0.06	0.0021	0.052	0.12
-10	0.09	0.00049	1.13
-15	0.06	0.0032	0.034	0.15
-15	0.09	0.0008	0.69
-20	0.06	0.0043	0.025	0.17
-20	0.09	0.0011	0.49

Table (1): result of simulation of ice freezing process.

Figure (1): ice mass flow rate as a function of ice block diameter and evaporating temperature.

Figure (2): maximum diameter of ice block as a function of evaporating temperature.

Figure (3): the average rate of heat flow per metre length by convection as a function of evaporating temperature and ice block diameter.

Conclusion

   The best evaporating temperature of ice freezing on a surface of pivotal evaporator is   (-20c), the decreasing rate of ice mass flow was 65% during the process ice block growth from (0.06M to 0.08 M ).At the same time using evaporating temperature (-5c), the decreasing rate of ice mass flow is 72 % during the same growth of ice block. The developed model and simulation method is flexible; hence they can be used as diagnostic tool to investigate the heat exchange modes of ice freezing at different water temperature, and different evaporator diameter. Such investigation increases the efficiency of ice freezing process by selection the optimum value of ice block diameter according to evaporator temperature, evaporator diameter and water temperature. 

References

[1] Chumak, G.and Chepurnenko, P.,1991, “ Refrigeration Plant “X–73, pp.344-362.
[2]  Arora, S., C. and Domcundwar, S., 1989,” Refrigeration and Air Conditioning,” Fourth Edition, pp.15.1-15.64.
[3]  Miekeef, M. and Miekeefa, I., 1977, “ Heat Transfer,” M69,pp.94-100.