Taylor series Temperature

All fluid properties are functions of space and time, namely p(x, y, z, t), p(x, y, z, t), T(x, y, z, t), and u(x, y, z, t) for the density, pressure, temperature, and velocity vector, respectively. The element under consideration is so small that fluid properties at the faces can be expressed accurately by the first two terms of a Taylor series expansion. For example, the pressure at the E and W faces, which are both at a distance l/26x from the element center, is expressed as... 

Although the equilibrium configuration of a molecule can usually be specified, at ordinary temperatures, all of the atoms undergo oscillatory motions. The forces between the atoms in the molecule are described by a Taylor series of the intramolecular potential function in the internal coordinates. This function can then be written in the form... 

The temperature dependence of the heat-conductivity for a certain film can be included as a truncated Taylor series ... 

The algebraic equations for the orthogonal collocation model consist of the axial boundary conditions along with the continuity equation solved at the interior collocation points and at the end of the bed. This latter equation is algebraic since the time derivative for the gas temperature can be replaced with the algebraic expression obtained from the energy balance for the gas. Of these, the boundary conditions for the mass balances and for the energy equation for the thermal well can be solved explicitly for the concentrations and thermal well temperatures at the axial boundary points as linear expressions of the conditions at the interior collocation points. The set of four boundary conditions for the gas and catalyst temperatures are coupled and are nonlinear due to the convective term in the inlet boundary condition for the gas phase. After a Taylor series expansion of this term around the steady-state inlet gas temperature, gas velocity, and inlet concentrations, the system of four equations is solved for the gas and catalyst temperatures at the boundary points. 

To obtain an expression for the deviation of the density p from the density p at the average temperature, p is expanded in a Taylor series about T ... 

The plot in Fig. 3.2 of the acid dissociation constant for acetic acid was calculated using equation 3.2-21 and the values of standard thermodynamic properties tabulated by Edsall and Wyman (1958). When equation 3.2-21 is not satisfactory, empirical functions representing ArC[ as a function of temperature can be used. Clark and Glew (1966) used Taylor series expansions of the enthalpy and the heat capacity to show the form that extensions of equation 3.2-21 should take up to terms in d3ArCp/dT3. 

But over wider ranges of temperature, ArS ° and AtH ° are functions of temperature. Clarke and Glew (1966) have used Taylor series expansions of the enthalpy... 

The nonlinearity in Eqs. (3.9)—(3.11) occurs in the product of variables and in the exponential temperature term. Expanding these nonlinear terms in a Taylor series and truncating after the first term give three linear ordinary differential equations ... 

The first domain corresponds to high-incident heat fluxes, where the pyrolysis temperature (TP) is attained very fast, thus t Application of the first-order Taylor Series expansion to Equation 3.13 around tp/tc —> 0 yields the following formulation for the pyrolysis time (lp) ... 

Taylor series as functions of experimental conditions. This is exactly analogous to the analysis of In r described previously except that, by means of a tentative model, the primary reaction rate dependence on concentrations, temperature, and other experimental factors has been eliminated. This permits the rate equations to be Integrated approximately correctly. 

To have an idea about the magnitude of the local discretization error, consider the Taylor serie.s expansion of the temperature at a specified nodal point m about time... 

Here, 0+ is the dimensionless temperature introduced by (1.37). With relatively small temperature differences a Taylor series... 

The equations (2.238) and (2.239) for the replacement of the derivatives with difference quotients can be derived using a Taylor series expansion of the temperature field around the point (Xi,tk), cf. [2.53] and [2.57]. It is also possible to derive the finite difference formula(2.240) from... 

One method to approximate the function AGq. is by using a Taylor series expansion around the temperature Tm, the melting temperature of pure a at pressure p. This yields... 

To gain some more insight into the effect of variations of temperature and/or fluid substrate attraction it is necessary to investigate the dependence of / (x A, Sa,o) on x. This becomes possible by considering small variations of X aroimd some rcforoncc value xq by expanding / (x A, s o) in a Taylor. series around this reference value xo > 0- Because of the definition of the variable X (see above) this may be considered either as an expansion in terms of or, alternatively, 1/T. More specifically, we write... 

In general, we are not only interested in solutions of Eq. (D.49) but also, more specifically, in those solutions satisfying Eq. (1.76a), which defines the chemical potential at coexistence between phases (i.e., morphologies) M" and for a given temperature T. To determine at a slightly different temperature T - T + ST, we expand u in a Taylor series around. some chemical potential /tj, say, so that... 

Truncating of the Taylor series at the second-order terms means that the second derivatives of the Gibbs free energy difference (ACp, Ak, Ad) do not change significantly with tempierature and pressure. If this assumption is not valid, an extended analysis is necessary, where the third order terms proportional to 7 , T p, T and p are involved. As a consequence, the form of the ellipse remains but it gets distorted [114], in particular at high temperatures and pressures. 

For small temperature differences, expanding both Tf and T into a Taylor series about a characteristic temperature To and subtracting,... 

At high temperature, the last term on the right-hand side of Equation (12) is significantly greater than zero and must be considered in the calculation. Criss (1991) expanded this term in a Taylor series, which, after canceling like terms, gives the following equation for diatomic molecules ... 

The desorption rate could then be expanded using a Taylor series about ZM (peak temperature) and eventually the half-width could be written as... 

Imagine the reactor is initially at this steady state and at t 0 we perturb the temperature and concentration by small amounts. We would like to know whether or not the system returns to the steady state after this initial condition perturbation. If so, we call the steady-state solution (asymptotically) stable. If not, we call the steady state unstable. Obviously we can solve numerically the nonlinear differential equations to answer this question, but then we answer the question on a case-by-case basis. By linearizing the nonlinear differential equations, we can gain further insight without resorting to full numerical solution. Consider the Taylor series expansion of the.nonlinear functions f, fz... 

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