Newtonian cooling

Here o is the density of the reactant mixture (perhaps measured in units of kg m-3), c the specific heat capacity (J K-1 kg-1), S the surface area, and X the surface heat transfer coefficient (Wm 2K 1). We have assumed a Newtonian cooling term for the transfer of heat to the surroundings. 

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,... 

If the temperature difference 0C between the heat bath and the inflow is greater than zero, we can have the opposite effect to Newtonian cooling, with a net flow of heat into the reactor through the walls. With his possibility, two more stationary-state patterns can be observed, giving a total of seven different forms—the same seven seen before in cubic autocatalysis with the additional uncatalysed step (the two new patterns then required negative values for the rate constant) or with reverse reactions included and c0 > ja0 ( 6.6). 

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.)...

Let us consider a simplified heat balance involving an exothermal reaction with zero-order kinetics. The heat release rate of the reaction q = f(T) varies as an exponential function of temperature. The second term of the heat balance, the heat removal by a cooling system qKX =f(T), with Newtonian cooling (Equation 2.18), varies linearly with temperature. The slope of this straight line is U-A and the intersection with the abscissa is the temperature of the cooling system Tc. This... 

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... 

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. 

The equation for heat flow from the environment to the sample or vice versa is given by the Newtonian cooling equation ... 

The rate of heat loss is assumed to be governed by Newtonian cool-... 

Convection, also known as Newtonian cooling (after Sir Isaac Newton), is a mechanism of heat transfer that occurs only in fluids. It involves the transfer of thermal energy by the mixing of fluids. The amount of convective heat transfer is a function of surface area in contact with the fluid, the temperature difference between the solid and the fluid, and the properties of the fluid. There are two types of convective heat flow — natural (or free) and forced. 

In 1939 Frank-Kamenetskii considered circumstances where Newtonian cooling was only an empirical approximation, and where the escape of heat was impeded internally by the thermal properties of the medium. (This will always be the case for a large enough system.) An internal temperature-distribution with a maximum at the middle results. For stability, this central temperature may not exceed a critical value. For a sphere with its surface at T the relationship is ... 

The simplest kinds of system are exemplified by a (heated) plane surface of a reactive material - like graphite in contact with a (heated) gas like air or oxygen. Products move out and reactants move in, and the surface recedes as solid combustion occurs. Chemically different but physically similar is the situation where the (heated) plane surface is not consumed, as when a solid catalyst accelerates an exothermic gas reaction. A Semenov-like treatment is quite appropriate with (i) Newtonian cooling representing heat flow and (ii) a parallel equation representing the flow of matter across a concentration step between surface layers and the bulk fluid. Ignition almost exactly... 

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... 

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)]. 

Autocatalysis with an unstable catalyst resembles non-isother-mal reaction under non-adiabatic operation. In each case the species responsible for feedback (B or the heat released) can be removed from the system by a route independent of the reactant A. These extra channels for removal are chemical decay and Newtonian cooling at the walls respectively. These circumstances lead to two independent variables. Only under stationary-state conditions is the concentration of A directly linked to the concentration of the catalyst or the temperature-excess. In our system we find... 

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. 

It is somehow similar to the case of heat transfer across the layer d, which can fall in between two optimal cases limited by 1/d and 7/cooling rate is essentially influenced by the heat transfer coefficient. A, and the thickness of cooled sample, d, and relatively less by its actual temperature, T. At the condition of ideal cooling, where we assume infinitely high coefficient of heat transfer, the cooling rate is proportional to l/(f, while for the Newtonian cooling controlled by the character of phase boundary, correlates to 1/d, only. In practice we may adopt the power relation = l/cf (where is an experimental constant lying again within 1< <2). 

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