Newtonian cooling equation

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

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]

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... 

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... 

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)]. 

The rates of heat transfer between the fermentation broth and the heat-transfer fluid (such as steam or cooling water flowing through the external jacket or the coil) can be estimated from the data provided in Chapter 5. For example, the film coefficient of heat transfer to or from the broth contained in a jacketed or coiled stirred-tank fermentor can be estimated using Equation 5.13. In the case of non-Newtonian liquids, the apparent viscosity, as defined by Equation 2.6, should be used. 

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. 

We consider the case of a horizontal liquid, layer confined between planes located at heights z = 0 and z = d, respectively. For simplicity we limit ourselves to the case d L, where L is either of the two horizontal scales of the layer. Later on we shall even restrict consideration to a (1 + 1)D two-dimensional geometry thus disregarding one of the two horizontal scales. The layer is assumed to be heated or cooled from below (Benard layer). In the simplest Newtonian and Boussinesquian approximations (to be defined more precisely) the evolution of the liquid layer is governed by the following balance equations ... 

Dimensionless analysis is a powerfiil tool in analyzing the transient heat transfer and flow processes accompanying melt flow in an injection mold or cooling in blown film,to quote a couple of examples. However, because of the nature of non-Newtonian polymer melt flow the dimensionless mmibers used to describe flow and heat transfer processes of Newtonian flnids have to be modified for polymer melts. This paper describes how an easily applicable equation for the cooling of melt in a spiral flow in injection molds has been derived on the basis of modified dimensionless numbers and verified by experiments. Analyzing the air gap dynamics in extrusion coating is another application of dimensional analysis. 

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