Newtonian cooling time

For the natural timescale we take the Newtonian cooling time tN... 

Here cp is the heat capacity (JK 1kg 1) and a the total density (kg m 3). If we divide the heat-balance equation throughout by cpa the Newtonian cooling time rN = cpa V/xS emerges naturally in the denominator of the last term, as does the group Q/cva which is related to the adiabatic temperature rise appropriate to the system, ATad = Qa0/cp[Pg.184]

Thus rres is the dimensionless residence time, as in the previous chapter, and tn is the dimensionless Newtonian cooling time. High values of rN correspond to slow heat transfer across the reactor walls, indicating well-insulated vessels which approach adiabatc operation as tn tends to infinity. Small values of tn correspond to systems which have fast heat transfer and hence which would be expected not to have great departures from isothermal operation. Finally, we need a measure of the activation energy... 

For a perfectly insulated reactor, with no heat loss through the walls, the Newtonian cooling time rN becomes infinite (because x - 0)- The mass- and heat-balance equations become... 

For finite, rather than infinite, values of the dimensionless Newtonian cooling time, the stationary-state condition is given by eqn (7.21). Thus, even with the exponential approximation, both R and L involve the residence time. The correspondence between tangency and ignition or extinction still holds,... 

In their analysis, which will form the basis of what follows here, Jorgensen and Aris chose to vary the Newtonian cooling time, keeping the residence time constant during any given experiment. Thus we may use tres as the timescale with which to make the rate equations dimensionless. The resulting forms, with the above simplifications, are... 

Fig. 13.20. The variation in the periodicity of the attractor wth the Newtonian cooling time rN for the consecutive exothermic reaction model in a CSTR. (Adapted and reprinted with permission from Jorgensen, D. V. and Aris, R. (1983). Chem. Eng. Sci., 38, 45-53.)...

Here, /r is a dimensionless measure of the initial reactant concentration Oo, y is the dimensionless concentration of X and k is the ratio of the Newtonian cooling time to the chemical reaction time at ambient temperature. 

Given ideal, well-stirred conditions, the heat release rate could be interpreted from (6.13) under non-stationary conditions, but accurate measurements of (dT/dt) would also be required. The rate of temperature change is always more important than the heat loss rate during the late stages of the development of ignition, because the chemical time-scale is much shorter than the Newtonian cooling time-scale. 

In normal circumstances, where operation is not strictly adiabatic, heat-losses may often be represented adequately by the value of the Newtonian cooling time, t = (aC V/hS). (For adiabatic operation, h 0 and -> oo). However, the same type of equation as before governs stationary states, save that B is replaced by where... 

Newtonian cooling time, arises in the same way Each of these represents the ratio of the rate of decay of B or of the heat evolved to the rate of their removal via the outflow. 

The heat-loss terms describe both the loss via the flow of gases leaving the reactor and via Newtonian cooling through the walls where f es is taken to be the average residence time of the reactor. is the ambient temperature, V the volume, 5 the reactor surface area and x the heat transfer coefficient. Cp is the heat capacity per unit volume which is assumed to be independent of temperature, and qj the exothermicity of the yth reaction step. 

CppV/xS, Newtonian cooling timescale tres mean residence time... 

This is a simplified method using only two points. A more accurate method is to use more points by applying the differential equation of Newtonian cooling, which expresses the variation with time of the temperature difference between reactor contents and cooling system [Eq. (25)]. 

Koelling et al. (70) conducted nonisothermal, pressurized gas-bubble Newtonian fluid-displacement experiments. The fluid used was PB H-300. It was injected into a capillary tube maintained at 60°C. The tube was then transferred in a different temperature bath at 0°C. The penetrating gas was then injected after different delay times, t. The longer the delay time, the deeper the cooling penetration thickness will be, since it is dependent on the Fourier number,... 

On the other hand, assuming the cooling mode of the fluid to be Newtonian, the quantity of heat transferred per unit time from the fluid, through the whole fluid surface, across the container walls, to the atmosphere, q2, is expressed as... 

This equation describes the time-temperature history of the solid object. The term c pV is often called the lumped thermal capacitance of the system. This type of analysis is often called the lumped capacity method or Newtonian heating or cooling method. 

Slow cooling due to poor crystaUizatimi causes extended cycle time (heat transfer) Deviation from Newtonian behaviour occurs at lower shear rates Melt fracture at higher shear rates (short chain molecules act as lubricants), increased expansion Narrow distribution results in better impact resistance... 

This phenomenon has been observed in both Newtonian and power law fluids (non-Newtonian) and is experimentally found to be closely related to polymer properties (molecular stmcrnre, molecular weight distribution, shear viscosity, relaxation time, power law index) as well as to the spinning variables, such as draw ratio, cooling rate, melt temperature, and die geometry, and so on. 

Thus a plot of the logarithm of the ratio of the instantaneous temperature difference to the initial temperature difference, as a fimction of time, should give a straight line, as far as cooling is Newtonian. The slope of this line is the thermal time constant of the reactor, from which the overall heat transfer coefficient U can be calculated. 

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