Equations heat flow

The balance equations for water vapor flows are similar to balance equations for contaminant flows, but in addition possible condensation and evaporation must be calculated. Also they must be considered in heat flow equations. 

Hicks (H6) and Frazer and Hicks (F3) considered the ignition model in which exothermic, exponentially temperature-dependent reactions occur within the solid phase. Assuming a uniformly mixed solid phase, the one-dimensional unsteady heat-flow equation relates the propellant temperature, depth from the surface, and time by the nonlinear equation ... 

At very high pressure, where the mean free path is short compared to the separation distance between surfaces, (a A jX)dP 1, and Eq. (101) reduces to the heat flow equation for moderate pressure gases given by Eq. (88). Although the derivation given here is not self-consistent when the mean free path is roughly the same as the separation distance, it is interesting to note that as the pressure becomes very small, (a A0/X)dP 1, and Eq. (101) reduces to the free molecular... 

Biot and Daughaday (B6) have improved an earlier application by Citron (C5) of the variational formulation given originally by Biot for the heat conduction problem which is exactly analogous to the classical dynamical scheme. In particular, a thermal potential V, a dissipation function D, and generalized thermal force Qi are defined which satisfy the Lagrangian heat flow equation... 

The flow of heat in a granular medium may be approximated by the classical heat-flow equations of mathematical physics. Assume a packing of particles contained in a rectangular box, one end of which is kept at a constant temperature. If T is the temperature at a distance x from the heat source, the flow of heat across an element of area A parallel to the heated surface will be — CA bTfbx), where C is the conductivity of the packing which is numerically equal to the quantity of heat flowing,... 

The electrical analogy may be used to solve more complex problems involving both series and parallel thermal resistances. A typical problem and its analogous electric circuit are shown in Fig. 2-2. The one-dimensional heat-flow equation for this type of problem may be written... 

Using the value of k in the heat flow equation above, we determine the temperature profile... 

The macroscopic phenomenological equation for heat flow is Fourier s law, by the mathematician Jean Baptiste Joseph Fourier (1768-1830). It appeared in his 1811 work, Theorie analytique de la chaleur (The analytic theory of heart). Fourier s theory of heat conduction entirely abandoned the caloric hypothesis, which had dominated eighteenth century ideas about heat. In Fourier s heat flow equation, the flow of heat (heat flux), q, is written as ... 

In the region of concentration in which the contributions of wall reactions arc regulated wholly or partially by diffusion, the concentrations of chain centers C will be governed by an equation very similar to the heat flow equation (XIV.2.1)... 

In other words, the value of Cp is calculated from the response to the modulation (Equation 4.5). If this is then multiplied by the linear component of heating rate, b, the heat capacity component of the underlying heat flow equation (Equation 4.6) is obtained. When this is subtracted from the underlying heat flow, the heat flow associated with the chemical reaction is obtained. In this way, these two different contributions to the heat flow are separated. 

This transformation arises from using equation (A8.12) instead of equation (A8.13) as the reference basis for heat flow. Equation (A8.15) is then replaced by ... 

In order to complete the analogy with the heat flow equation, it is necessary to choose the integration constant a, in such a way that t is everywhere positive and 0 = 0 coincides with t = 0. This happens at a, = 2/3, so that explicitly... 

We will also have heat flow equations similar to those for the TS-BR and TS-PFR reactors, but with the addition of a new term to account for cool/hot inlet feed mixing with hot/cool reactant inside the reactor. To find an expression for this term, consider, over a short time dt, a volume of fluid fbdt altering the reactor at temperature TE It expands to volume Syf) inside the reactor, and mixes with the remaining volume V- 5v fb dt which is at temperature T. The mixture then has a resulting temperature of. 

We use the same approach for DSC as we did for DTA. We start with the thermal heat flow equation which is similar to Ohm s Law, vis-... 

Using conventional heat flow equations, and assuming that the total heat is generated at the interface, the temperature distribution in the two solids can be... 

Thus, it is possible to obtain the global heat flow out of the reactor by adding up the resistances contributed by each of the layers and using the resulting total resistance in the heat flow equation. 

By substituting these values into the heat flow equation, the global heat flow from the inner reactor conducted through the various layer and convected out of the system through natural convection is found ... 

Conservation of energy dictates that the heat will pass through each of the layers. Thus, it is possible to calculate the interfacial temperature between each layer by applying the same heat flow equation, but focusing around a single thermal resistance. For example, in the case of the alumina refractory layer ... 

Including also the heat capacities of the sample and reference platforms, Cs, into the model leads to the bottom sketch in Fig. A. 11.2. The heat-flow rate into the sample needs now a four-term heat-flow-equation making use of a second temperature difference AT = Tj - T, besides AT = T - Tji... 

The general heat flow equation (equation 2.3) describing TMDSC assumes that in regions where no kinetic phenomena occur the sample responds... 

An extensive analysis of the sawtooth modulation brought a number of interesting results. Mathematically, it could be shown that if there were no temperature gradients within the sample and if all other lags and gradients could be assessed with the Fourier heat-flow equation, Eq. (11) does allow the calculation of the precise heat capacities [33]. Temperature gradients are, however, almost impossible to avoid. Especially in the power-compensated calorimeter, the temperature sensor is much closer to the heater than the sample and cannot avoid gradients. The empirical solution to this problem was to modify Eq. (11) as follows [34] ... 

A rigorous treatment of multidimensional heat flow leads to a time-dependent heat flow equation in a conducting medium, which in Cartesian coordinates, and for Qy(W/m ) being the internal heat... 

So, one way in which to solve a diffusion equation is to solve instead the corresponding heat flow equation having the same boimdary conditions, and replace T with c, and k with D, You can find solutions to diffusion problems in books of either diffusion or heat conduction and In problems having the given boundary conditions. For the present problem, see Equation (2.6) in [1]. The solution to this one-dimensional problem is... 

The heat flow equation for the change between solid and liquid is like the equation for the liquid-to-gas change ... 

Douglas J Jr, GaUie TM Jr (1955) Variable time steps in the solution of the heat flow equation by a difference equation. Proc Am Math Soc 6 787-793... 

Heat flow equations are derived using a model insulation system given in Fig. AID.l. 

Both heat flow across and that generated in the element are summed up and the total heat flow equation is represented by... 

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