Differential Equations of Continuity

In Sections 2.6,2.7, and 2.8 overall mass, energy, and momentum balances allowed us to solve many elementary problems on fluid flow. These balances were done on an arbitrary finite volume sometimes called a control volume. In these total energy, mechanical energy, and momentum balances, we only needed to know the state of the inlet and outlet streams and the exchanges with the surroundings. 

These overall balances were powerful tools in solving various flow problems because they did not require knowledge of what goes on inside the finite control volume. Also, in the simple shell momentum balances made in Section 2.9, expressions were obtained for 

For nonisothermal systems a general differential equation of conservation of energy will be considered in Chapter 5. Also in Chapter 7 a general differential equation of continuity for a binary mixture will be derived. The differential-momentum-balance equation to be derived is based on Newton s second law and allows us to determine the way velocity varies with position and time and the pressure drop in laminar flow. The equation of momentum balance can be used for turbulent flow with certain modifications. 

Often these conservation equations are called equations of change, since they describe the variations in the properties of the fluid with respect to position and time. Before we derive these equations, a brief review of the different types of derivatives with respect to time which occur in these equations and a brief description of vector notation will be given. 

Partial time derivative. Various types of time derivatives are used in the derivations to follow. The most common type of derivative is the partial time derivative. For example, suppose that we are interested in the mass concentration or density p inkg/m in a flowing stream as a function of position x, y, z and time t. The partial time derivative of p is dp/dt. This is the local change of density with time at a fixed point x, y, and z. 

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As we do for all mass transfer problems, we must satisfy the differential equation of continuity for each species as well as the differential momentum balance. Since we are dealing with a porous medium having a complex and normally unknown geometry, we choose to work in terms of the local volume averaged forms of these relations. Reviews of local volume averaging are available elsewhere (23-25). 

In this discussion overall or macroscopic mass balances were made because we wish to describe these balances from outside the enclosure. In this section on overall mass balances, some of the equations presented may have seemed quite obvious. However, the purpose was to develop the methods which should be helpful in the next sections. Overall balances will also be made on energy and momentum in the next sections. These overall balances do not tell us the details of what happens inside. However, in Section 2.9 a shell momentum balance will be made to obtain these details, which will give us the velocity distribution and pressure drop. To further study these details of the processes occurring inside the enclosure, differential balances rather than shell balances can be written and these are discussed in other later Sections 3.6 to 3.9 on differential equations of continuity and momentum transfer. Sections 5.6 and 5.7 on differential equations of energy change and boundary-layer flow, and Section 7.5B on differential equations of continuity for a binary mixture. 

B Differential Equations of Continuity and Motion for Flow between Parallel Plates... 

Figure 3.8-3. Horizontal flow in a lube in Example 3.8-3. Sec. 3.8 Use of Differential Equations of Continuity and Motion...

In Sections 3.6 and 3.7 we derived a differential equation of continuity and a differential equation of momentum transfer for a pure fluid. These equations were derived because overall mass, energy, and momentum balances made on a finite volume in the earlier parts of Chapter 2 did not tell us what goes on inside a control volume. In the overall balances performed, a new balance was made for each new system studied. However, it is often easier to start with the differential equations of continuity and momentum transfer in general form and then to simplify the equations by discarding unneeded terms for each specific problem. 

In Section 3.8 we used the differential equations of continuity and of motion to set up fluid flow problems. We did this by discarding the terms that are zero or near zero and using the remaining equations to solve for the velocity and pressure distributions. This was done instead of making new mass and momentum balances for each new situation. In a similar manner, to solve problems of heat transfer, the differential equations of... 

No assumption has been made as to continuity, in general, but it will now be assumed that all functions have continuous derivatives of order n + 1. Then the t satisfy a linear ordinary differential equation of order n + 1, which can be written in the form... 

Based on this configuration, the reformer and combustor are modeled with partial differential equations. Since the thickness of the plates is relatively small, only the flow direction is considered. Using the equation of continuity, the component mass balances are constructed and the energy balance considering with heat loss and momentum balance are established as follows. 

The KG equation is Lorentz invariant, as required, but presents some other problems. Unlike Schrodinger s equation the KG equation is a second order differential equation with respect to time. This means that its solutions are specified only after an initial condition on bothand d /dt has been given. However, in contrast to k, d /dt has no direct physical interpretation [61]. Should the KG equation be used to define an equation of continuity, as was done with Schrodinger s equation (4), it is found to be satisfied by... 

The three-dimensional transport equation for inert pollutant dispersion results from timesmoothing the equation of continuity of the emitted substance. In Cartesian coordinates the distribution of a pullutant is given by the partial differential equation of second order for the concentration C(x, y, z, t) 111 ... 

Eqs. (2.14), (2.30), and (2.31) with Eq. (2.9) can be solved simultaneously without simplification. Since the analytical solution of the preceding simultaneous differential equations are not possible, we need to solve them numerically by using a computer. Among many software packages that solve simultaneous differential equations, Advanced Continuous Simulation Language (ACSL, 1975) is very powerful and easy to use. 

The LGA is a variant of a cellular automaton, introduced as an alternative numerical approximation to the partial differential equation of Navier-Stokes and the continuity equations, whose analytical solution leads to the macroscopic approach of fluid dynamics. The microscopic behavior of the LGA has been shown to be very close to the Navier-Stokes (N-S) equations for incompressible fluids at the macroscopic level. 

Another commonly used model is based on the general differential balance of mass and momentum [Burgers, 1948]. Consider a steady, incompressible, and axially symmetric flow in which the body forces are negligible. In cylindrical coordinates, the equation of continuity of the fluid can be given as... 

The Equation of Continuity by Differential Mass Balance Derive the equation of continuity in cylindrical coordinates by making a mass balance over the differential volume Ar(rA6)Az. 

Process Transfer Function Models In continuous time, the dynamic behaviour of an ideal continuous flow stirred-tank reactor can be modelled (after linearization of any nonlinear kinetic expressions about a steady-state) by a first order ordinary differential equation of the form... 

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