Partial differentiation changing independent variables

One of the procedures found useful in solving partial differential equations is the so-called combination of variables method, or similarity transform. The strategy here is to reduce a partial differential equation to an ordinary one by judicious combination of independent variables. Considerable care must be given to changing independent variables. 

All the coefficients will, in general, be functions of both independent variables, and since we know that the heat absorbed depends on the path of change, it follows that the coefficients are not, in general, partial derivatives of a function of the two independent variables, for SQ would then be a perfect differential (cf. H. M., 115). 

To change the independent variables w /, mj, m"9 m2 to s, s" we have the relations (11) and the following, derived from them by partial differentiation ... 

The quantities dX and d Y are called differentials, the coefficients in front of dX and dT are called partial derivatives,11 and dZ is referred to as a total differential because it gives the total change in Z arising from changes in both X and Y. If Z were to depend upon additional variables, additional terms would be included in equation (A 1.1) to represent the changes in Z arising from changes in those variables. For much of our discussion, two variables describe the processes of interest, and therefore, we will limit our discussion to two independent variables, with the exception of the description of Pfaffian differentials in... 

Equation 2.2-8 indicates that the internal energy U of the system can be taken to be a function of entropy S, volume V, and amounts nt because these independent properties appear as differentials in equation 2.2-8 note that these are all extensive variables. This is summarized by writing U(S, V, n ). The independent variables in parentheses are called the natural variables of U. Natural variables are very important because when a thermodynamic potential can be determined as a function of its natural variables, all of the other thermodynamic properties of the system can be calculated by taking partial derivatives. The natural variables are also used in expressing the criteria of spontaneous change and equilibrium For a one-phase system involving PV work, (df/) 0 at constant S, V, and ,. ... 

The partial differential Equation 27 can be simplified to an ordinary differential equation in the independent variable z, by the following approximation For typical FCC catalysts and feedstocks, c A - 3 x CpC, and for the MAT test, [C/O] is 3 (Table I). Hence, if we assume that the catalyst and oil temperatures are identical (no heat transfer resistance between oil and catalyst), then as a first approximation, the change in the energy content of the oil and of the catalyst are roughly the same over the length of the run. Thus, the two terms on the left hand side of Equation 27 are approximately the same magnitude. Therefore, the time derivative of T can be lumped with the distance derivative. The right hand side of Equation 27 is divided by 2, and it becomes ... 

If z depends on more than two independent variables, an additional term is added to (A.2.1) for each additional variable. The total differential (A.2.1) can be used to form other partial derivatives for example, to express how z responds to changes in x with another quantity w held fixed, we use (A.2.1) to write... 

Now 7 is a function of state and thus forms an exact differential. It is also known experimentally to bo a smooth function of the variables 8y V and n, , except at points of phase change. If we disregard such points, the second partial differential of U with respect to any pair of variables is independent of tho order of differentiation. 

In order to compute the change of, say, the enthalpy it is desirable to use temperature and pressure as the independent variables and also to express the various partial differential coefficients in terms of experimental magnitudes. This may be illustrated as follows. 

A dependent variable is a function of the independent variables. The total differential of a dependent variable is an expression for the infinitesinial change of the variable in terms of the infinitesimal changes of the independent variables. As explained in Sec. F.2 of Appendix F, the expression can be written as a sum of terms, one for each independent variable. Each term is the product of a partial derivative with respect to one of the independent variables and the infinitesimal change of that independent variable. For example, if the system has two independent variables, and we take these to be T and V, the expression for the total differential of the pressme is... 

Second, the balance is taken over an incremental space element, Ac, Ar, or AV. The mass balance equation is then divided by these quantities and the increments allowed to go to zero. This reduces e difference quotients to derivatives and the mass balance now applies to an infinitesimal point in space. We speak in this case of a "difference" or "differential" balance, or alternatively of a "microscopic" or "shell" balance. Such balances arise whenever a variable such as concentration undergoes changes in space. They occur in all systems that fall in the category of the device we termed a 1-D pipe (Figure 2.1b). When the system does not vary with time, i.e., is at steady state, we obtain an ODE. When variations with time do occur, the result is a partial differential equation (PDE) because we are now dealing with two independent variables. Finally, if we discard the simple 1-D pipe for a multidimensional model, the result is again a PDE. 

The differential of a function of several variables (an exact differential) has one term for each variable, consisting of a partial derivative times the differential of the independent variable. This differential form delivers the value of an infinitesimal change in the function produced by infinitesimal changes in the independent variables. 

This equation is simply a quite general first-order Taylor series. It states that the differential change of any variable is the sum of the product of its partial derivatives times the differential changes in the independent variables. It is slightly modified from the Taylor series because we have held Tand P constant, thus eliminating the terms in dT and dP. However, we see from it that the derivatives that appear on the right are the partial molar derivatives of Y. For example, if we let T be volume, then... 

Apply thermodynamics to mixtures. Write the differential for any extensive property, dK, in terms of m + 2 independent variables, where m is the number of species in the mixture. Define and find values for pure species properties, total solution properties, partial molar properties, and property changes of mixing. 

If we choose temperature, pressure, and mole fraction of species i as independent variables, the change in chemical potential divided by temperature can be related by the following partial differentials ... 

If equation (2.51) is the total differential for as a function of two variables, 1 and 2, we can expect that its partial derivatives (d E/d Zi) and (<9 /c> 2)5 can be expressed as functions of only those two variables. That is, — ( , 2). Thus, derivatives of (<9 /<9 ) and (d E/d Zi)- with respect to variables other than 1 and 2 should be zero. As we consider the implications of this statement, it is important to note that a change can be made independently in the r variable of one subsystem without affecting that of the other, but a change in 0 will affect both subsystems (since 0 is the same in both subsystems). Therefore, we must consider the implications for c and 0 separately in the analysis that follows. 

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