Heat generation, equation

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... 

AH the reduction reactions are endothermic, regardless of the reductant used. The heat for these reactions, along with the requirements for the sensible heats of the hot metal and slag, and heat losses through the furnace shell, is provided by the heat generated from equation 1 plus the sensible heat of the hot blast. 

The burning of coke in the regenerator provides the heat to satisfy the FCCU heat balance requirements as shown in equation 1. The heat released from the burning of coke comes from the reaction of carbon and hydrogen to form carbon monoxide, carbon dioxide, and water. The heat generated from burning coke thus depends on the hydrogen content of the coke and the relative amounts of carbon that bum to CO and CO2, respectively. 

Several Bodies in Series with Heat Generation The simple Fourier type of equation indicated by Eq. (5-15) may not be used when heat generation occurs in one of the bodies in the series. In this case, Eq. (5-5c), (5-5Z ), or (5-5c) must be solved with appropriate boundaiy conditions. 

In such a condition, if the heat generated in the windings raises the temperature of the windings by 6 above the temperature, the motor was operating just before stalling. Then by a differential form of the heat equation ... 

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

The above equations for heat transfer apply when there is no heat generation or absorption during the reaction, and the temperature difference between the solid and the gas phase can be simply defined tliroughout the reaction by a single value. Normally this is not the case, and due to the heat of the reaction(s) which occur tlrere will be a change in the average temperature with time. Furthermore, in tire case where a chemical reaction, such as the reduction of an oxide, occurs during the ascent of tire gas in the reactor, the heat transfer coefficient of the gas will vary with tire composition of tire gas phase. 

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... 

Because of this heat generation, when adsorption takes place in a fixed bed with a gas phase flowing through the bed, the adsorption becomes a non-isothermal, non-adiabatic, non-equilibrium time and position dependent process. The following set of equations defines the mass and energy balances for this dynamic adsorption system [30,31] ... 

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. 

Assume a FT boiler of, say, 500 HP, having 2,500 sq ft of heat generating surfaces (and 225 sq ft of internal shell below the waterline, which becomes hot and to which scale can adhere). If we also assume that the boiler has a uniform deposit of scale and corrosion debris on all waterside surfaces to an eggshell thickness (31 mil), then the total mass of dirt equates to ... 

Setting Equation (5.32) equal to Equation (5.33) gives the general heat balance for a steady-state system. Figure 5.6(c) shows the superposition of the heat generation and removal curves. The intersection points are steady states. There are three in the illustrated case, but Figure 5.6(d) illustrates cases that have only one steady state. 

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 numerical method of solving the model using computer tools does not require the explicit form of the differential equation to be used except to understand the terms which need to be entered into the program. The heater and the barrel were modeled as layers of materials with varying thermal characteristics. The energy supplied was represented as a heat generation term qg) in a resistance wire material. Equation 1... 

C06-0093. Solid urea, (NH2)2 CO, bums to give CO2, N2, and liquid H2 O. Its heat of combustion is -632.2 kJ/mol. (a) Write the balanced combustion equation, (b) Calculate the heat generated per mole of H2 O formed, (c) Using this heat of combustion and the appropriate thermod3uamic data, determine the heat of formation of urea. 

Regions of stable and unstable operation determined by numerical simulation of mass and heat balances equations first- and second-order, autocatalytic, and product-inhibited kinetics graphically presented boundaries in co-ordinates in practice. safe operation if l/5e>2. Equality of heat generation and heat removal rates Semenov approach modified for first-order kinetics. 

Note that this includes terms for the rates of heat generation of both reactions. As shown in Sec. 1.2.5, the heat balance equation is equivalent to... 

In a steady-state situation all net heat generated is emitted to space. Therefore, by adding the calculated geothermal mean heat flow (0.068 W m-2) and the total thermal pollution (0.020 W m-2), the total net OLR becomes 0.088 W m-2. Equation (3) then gives that this net OLR requires a SST of 17.8 °C and a LAT of 12.0 °C resulting in a future global mean temperature of 16.1 °C. 

The total heat flow (Q) at the film surface is equal to the mass flow rate (W k) times the heat of condensation (AH). That is, the heat generated by condensation at the surface must be equal to the heat transported away by conduction and radiation for steady state to be achieved. In mathematical terms this results in the equation. 

The heat generation function can be determined from equation 12.3.108. 

In order to find points of equal degrees of conversion (or equal Q-values) in Figure 2.18, van Geel [115] developed the method to evaluate kinetic data from the so-called isoconversion lines. A heat generating substance that follows Equation (2-11), when stored under isothermal conditions at different temperatures has generated an equal amount of heat (Q) when the product of t exp(-Ea/RT) has the same value. Thus, for two heat generation/time curves measured at Ti and T2, the same amount of heat (Q) has been generated, and thus the conversion is equal when ... 

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