Heat Transfer Within A Jacketed Reactor System
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Modeling of heat transfer within a jacketed reactor requires basic knowledge on process heat transfer; reactor design etc. literature review sum up the fundamental on energy balance, method of overall heat transfer coefficient determination and basic understanding of crystallization. These are the basic methods which allow engineers to predict more accurate capabilities during chemical process as well as timing on the process.
Heat transfer is important in agitated vessels due to fluid temperature is the most significant factor for controlling the outcome of chemical, biochemical and pharmaceutical processes. 
Jacketed agitated vessels for heating and cooling are commonly used in vary types of process applications. Engineers should have working knowledge of how heat transfer and temperature control principles applied to such vessels. Cooling or heating agitated liquid in vessels is a basic technological operation on the chemical, biochemical, pharmaceutical, food and processing industries. The cooling or heating rate depends on how the heat is supplied or removed, the mixing intensity and many other parameters.  The temperature needs to be controlled precisely at its desired to meet the requirement of downstream operations. Hence a mathematical model is essential which can predict temperatures accurately.
The rate of heat transfer to or from an agitated liquid mass in a vessel is a function of the physical properties of that liquid and of the heating or cooling medium, the vessel geometry, and the degree of agitation.  Other factors which may affect the rate of heat transfer include type and size of the agitator and agitator location in the vessel. Most of the jacketed agitated vessels are used as reactor, thus chemical reactions with exothermic or endothermic effects must be taken into account as well. In a vessel containing an agitated liquid, heat transfer takes place mainly through conduction and forced convection, as it does in heat exchangers. 
Crystallization is a unit operation for separation and production of pure solid materials with desired properties. To develop a batch cooling crystallization process, various operation strategies need to be investigated in relation to seeding, cooling, mixing, fines dissolution, and so forth.  In commercial scale process, the reactor size grows larger. In this situation, various problems like ancillary nucleation, attrition, breakage, agglomeration, and dead zone may become severer in relation to the increasing inhomogeneities in the solution temperature and hydrodynamics.
Modeling of reactors is useful for analyzing data, estimating performance, reactor scale-up, simulating start-up and shut down behavior, and control.  Uncertainties such as scale-up options, explosion hazards, runaway reactions, environmental emissions, reactor internals etc, may be explored through modeling.  A key aspect of modeling is to derive the appropriate momentum, mass or energy conservation equations for the reactor.
One typical application in heat transfer with batch operation is heating the process fluid in reactor, maintaining temperature during the reaction period and cooling the product after reaction complete. 
The overall thermal energy balance includes the heat entering the system, heat leaving the system, heat accumulation and heat loss. The equation can be written as:
In batch process, there is no liquid or fluid entering or leaving the system. If the system is assumed to be perfectly insulated, the energy balance equation can be simplified in: 
By integration of both sides:
For a batch manufacturing process, heat transfer in an agitated vessel is used to design a suitable process or reaction. It is necessary to calculate the time to heat or cool a batch or the cooling capacity required to hold an exothermic or endothermic reaction at constant temperature.  The technique is to develop an expression which is relating time for heating or cooling agitated batches to coil or jacket area, heat-transfer coefficient, and the heat capacity of the vessel contents.  By rearranging the energy balancing equation, the relevant equation to calculate time is as follow:
This equation only can be used in where the utility fluid temperature remains constant or the fluid temperature difference between inlet and outlet is not greater than 10% of the log mean temperature difference between the average temperature of the jacket and the temperature of the vessel's content.  Precisely, for heating and cooling condition, this equation must be represented in separately:
If the situation is greater than 10% of the log mean temperature difference, the apply equation will be:
W = the mass flow rate through the jacket,
C = the specific heat of the fluid in the jacket
Assumptions are made for solving energy balance equation  
U is constant for the process and over the entire surface
Liquid flow rates are constant
Specific heats are constant for the process
The heating or cooling medium has a constant inlet temperature
Agitation produces a uniform batch fluid temperature
No partial phase changes occurs
Heat losses are negligible
Agitated vessel heat transfer coefficient
Process side heat transfer coefficient can be determined by speed and agitator type. For low viscosity fluids, high-speed turbine type agitators will provide good performance. For high viscosity fluids and non-newtonian fluids, larger diameter agitators will be more suitable. 
Various types of agitators are used for mixing and blending as well as to promote heat transfer in vessels. The correlations used to estimate the heat transfer coefficient to the vessel wall. 
For agitated vessels:
hv = heat transfer coefficient to vessel wall or coil, Wm-2â„ƒ-1
D = agitator diameter, m
N = agitator, speed, rps (revolutions per second)
ρ = liquid density, kg/m3
kf = liquid thermal conductivity, Wm-1â„ƒ-1
Cp = liquid specific heat capacity, J Kg-1â„ƒ-1
μ = liquid viscosity, Nm-2s.
The values of constant C and the indices a, b and c depend on the type of agitator the use of baffles, and whether the transfer is to the vessel wall or to coils. Some typical correlations are given below: 
Flat blade disc turbine, baffled or unbaffled vessel, transfer to vessel wall, Re < 400:
Flat blade disc turbine, baffled vessel, transfer to vessel wall, Re> 400:
Overall heat transfer coefficient
Most utility and process fluid will foul the heat transfer surfaces in an exchanger to a greater or lesser extent. The deposited material will normally have a relatively low thermal conductivity and will reduce the overall coefficient.
Fouling factors usually are considered in determining the Overall heat transfer coefficient U. The overall heat transfer coefficient is calculated in this way:
Where α and αs are the heat transfer coefficients for the process and utility side respectively. On the utility side, fouling resistance 1/αf can be found from local experience or from Kern (1950). 
Heat transfer utility fluid
Syltherm 800 is a silicone heat transfer fluid. It is a highly stable, long-lasting silicone fluid designed for high temperature liquid phase operation. It exhibits low potential for fouling and can often remain in service for 10 years or more. The recommended using temperature range is. 
Crystallization occurs with generating a sufficient level of supersaturation. The method of generation of supersaturation is to provide heat transfer, which is used in cooling and evaporative crystallization processes. There are two essential steps for crystallization: nucleation and crystal growth.
The problems of scale-up in crystallization process can be classified into induced, hydrodynamically induced, and mixes. For example, attrition, breakage, and agglomeration are related to solution mixing and are investigated from the hydrodynamic point of view. On the other hand, ancillary nucleation is caused by increased temperature gradient within the solution together with seed particles generated by attrition or fluid shear and can be considered as an example where the thermal and hydrodynamic effects are mixed. To improve the hydrodynamics deterioration during the scale-up, impeller type, agitation power, and baffle or draft tube design2,8,9 can be modified or newly designed as required. The thermal aspect improvement is performed by the heat transfer enhancement, but the remedies are limited because the heat transfer area to volume ratio decreases inevitably during the scale-up unless other techniques such as vacuum or evaporative crystallization is introduced.
Calculation of time to heat or cool a fixed amount of liquid inside a batch reactor usually assume the process and utility heat capacity and the overall heat transfer coefficient to be constant throughout the calculations.
Equations (liquid in jacket) heat input to reactor at T = heat loss by utility liquid with inlet temperature T1 and outlet temperature T2
Rearrange the equation to solve unknown jacket outlet temperature T2
The rate of temperature change of the liquid inside the vessel is given by
Solving the above two equations to get process temperature as a function of time
Finally, solving for time t where T = Tf
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