Heat generation rate equations

Temperature gradient normal to flow. In exothermic reactions, the heat generation rate is q=(-AHr)r. This must be removed to maintain steady-state. For endothermic reactions this much heat must be added. Here the equations deal with exothermic reactions as examples. A criterion can be derived for the temperature difference needed for heat transfer from the catalyst particles to the reacting, flowing fluid. For this, inside heat balance can be measured (Berty 1974) directly, with Pt resistance thermometers. Since this is expensive and complicated, here again the heat generation rate is calculated from the rate of reaction that is derived from the outside material balance, and multiplied by the heat of reaction. 

In the simplest case of one-dimensional steady flow in the x direction, there is a parallel between Eourier s law for heat flowrate and Ohm s law for charge flowrate (i.e., electrical current). Eor three-dimensional steady-state, potential and temperature distributions are both governed by Laplace s equation. The right-hand terms in Poisson s equation are (.Qy/e) = (volumetric charge density/permittivity) and (Qp // ) = (volumetric heat generation rate/thermal conductivity). The respective units of these terms are (V m ) and (K m ). Representations of isopotential and isothermal surfaces are known respectively as potential or temperature fields. Lines of constant potential gradient ( electric field lines ) normal to isopotential surfaces are similar to lines of constant temperature gradient ( lines of flow ) normal to... 

If heat ean be removed as fast as it is generated by the reaetion, the reaetion ean be kept under eontrol. Under steady state operating eonditions, the heat transfer rate will equal the generation rate (see Figure 6-26). If the heat removal rate Qj. is less than the heat generation rate Qg (e.g., a eondition that may oeeur beeause of a eooling water pump failure), a temperature rise in the reaetor is experieneed. The net rate of heating of the reaetor eontent is the differenee between Equations 12-44 and 12-45. 

Equation (8.29) provides no guarantee of stability. It is a necessary condition for stability that is imposed by the discretization scheme. Practical experience indicates that it is usually a sufficient condition as well, but exceptions exist when reaction rates (or heat-generation rates) become very high, as in regions near thermal runaway. There is a second, physical stability criterion that prevents excessively large changes in concentration or temperature. For example. An, the calculated change in the concentration of a component that is consumed by the reaction, must be smaller than a itself Thus, there are two stability conditions imposed on Az numerical stability and physical stability. Violations of either stability criterion are usually easy to detect. The calculation blows up. Example 8.8 shows what happens when the numerical stability limit is violated. 

The RC1 reactor system temperature control can be operated in three different modes isothermal (temperature of the reactor contents is constant), isoperibolic (temperature of the jacket is constant), or adiabatic (reactor contents temperature equals the jacket temperature). Critical operational parameters can then be evaluated under conditions comparable to those used in practice on a large scale, and relationships can be made relative to enthalpies of reaction, reaction rate constants, product purity, and physical properties. Such information is meaningful provided effective heat transfer exists. The heat generation rate, qr, resulting from the chemical reactions and/or physical characteristic changes of the reactor contents, is obtained from the transferred and accumulated heats as represented by Equation (3-17) ... 

Equation (31) shows an instanee of a temperature profile function which incorporates the thickness of a speeifie polymer tpoiy, the width of the heater eleetrode Wheat, and the heat generation rate as parameter, and constants c and Cl are fitted from caleulated temperature profiles,... 

These equations calculate the temperature and conversion profiles in a polytropic tubular reactor. The term (a) represents the heat generation rate by the reaction and the term (b) the heat removal rate by the heat exchange system. This equation is similar to Equation 5.2, obtained for the batch reactor. Moreover, since the... 

Following the procedure used with the one-dimensional FEM model and using the constant strain triangle element developed in the previous section, we can now formulate the finite element equations for a transient conduction problem with internal heat generation rate per unit volume of Q. The governing equation is given by... 

Tire temperature and mechanical properties of cords. The heat generation from cord and rubber under cyclic straining In rolling naturally causes the tire temperature to rise. Dynamics of this temperature rise can be expressed by the following heat transfer equation with the heat generation rate terms for cord and rubber. (3)... 

In equation (1), Cp, T, t, K, v2, Qj., v< , v are density, heat capacity, temperature, time, thermal conductivity, Laplaclan operator, heat generation rates of cord and rubber, volume fractions of cord and rubber, respectively. The equation can be solved numerically by use of either the finite difference approximation or the finite element method. The solution of the equation with related boundary conditions provide the temperature profile In the tire wall cross section. Examples of such solutions are shown In Figure 2. The procedure of obtaining such solutions Is outlined In Appendix I. 

Table 1. Comparison of the measured rolling resistance and the rolling resistance which was calculated from the total heat generation rate by use of the relationship given by equation (2)...

In order to determine the temperature profile in the tire cross section shown In Figure 2, the heat transfer equation (Equation 1) has to be solved with proper boundary conditions, material properties and the heat generation rate of cord and rubber appearing In Equation (1). In this pendlx, we give a brief description of the procedures Involved In this. 

Inserting Equation 3.246 for x f in the above equation, we get the equation for the heat generation rate Qg as a function of the reactor temperature attained at steady state. 

In practice, equation (6.10) tends to overestimate the maximum heat generation rate because the peak-to-average flux is usually less than the value... 

Using equation (6.35) to evaluate the derivative, and equating the heat conduction rate to the heat generation rate in the fuel, we find for the cladding surface temperature... 

A left part of the obtained correlation depends linearly on the temperature T. Its value is proportional to the cooling rate caused by a hot (T > Tq) airflow out of the reactor. A right part corresponds to the heat generation rate in the reactor due to the reaction exothermicity. It is a non-linear function of temperature. If one plots the temperature dependences of the equation right and left parts, one will see their interception in points corresponding to the calculated stationary temperatures and concentrations (Fig. 3.31). 

Assuming that the heat generation is due to the difference between the operating voltage and the maximum possible output voltage, the heat generation rate is given by Equation 4.101 ... 

Heat generation rate is given in Equation 4.101 for a fuel cell stack as... 

In the Couette flow inside a cone-and-plate viscometer the circumferential velocity at any given radial position is approximately a linear function of the vertical coordinate. Therefore the shear rate corresponding to this component is almost constant. The heat generation term in Equation (5.25) is hence nearly constant. Furthermore, in uniform Couette regime the convection term is also zero and all of the heat transfer is due to conduction. For very large conductivity coefficients the heat conduction will be very fast and the temperature profile will... 

Consider a negative sequence component ol 40% of the rated current. Then the maximum heat generated as in equation (12.4)... 

Chemical reactions obey the rules of chemical kinetics (see Chapter 2) and chemical thermodynamics, if they occur slowly and do not exhibit a significant heat of reaction in the homogeneous system (microkinetics). Thermodynamics, as reviewed in Chapter 3, has an essential role in the scale-up of reactors. It shows the form that rate equations must take in the limiting case where a reaction has attained equilibrium. Consistency is required thermodynamically before a rate equation achieves success over tlie entire range of conversion. Generally, chemical reactions do not depend on the theory of similarity rules. However, most industrial reactions occur under heterogeneous systems (e.g., liquid/solid, gas/solid, liquid/gas, and liquid/liquid), thereby generating enormous heat of reaction. Therefore, mass and heat transfer processes (macrokinetics) that are scale-dependent often accompany the chemical reaction. The path of such chemical reactions will be... 

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