Differential Equation of Energy Change

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 Chapter 4 on steady-state heat transfer and Chapter 5 on unsteady-state heat transfer new overall energy balances were made on a finite control volume for each new situation. To advance further in our study of heat or energy transfer in flow and nonflow systems we must use a differential volume to investigate in greater detail what goes on inside this volume. The balance will be made on a single phase and the boundary conditions at the phase boundary will be used for integration. 

As in the derivation of the differential equation of momentum transfer, we write a balance on an element of volume of size Ax, Ay, and Az which is stationary. We then write the law of conservation of energy, which is really the first law of thermodynamics for the fluid in this volume element at any time. The following is the same as Eq. (2.7-7) for a control volume given in Section 2.7. 

As in momentum transfer, the transfer of energy into and out of the volume element is by convection and molecular transport or conduction. There are two kinds of energy being transferred. The first is internal energy U in J/kg(btu/lb ,) or any other set of units. 

The total energy coming in by convection in the x direction at x minus that leaving at X + Ax is 

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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. 

In Chapter 5 the conservation-of-energy equations (2.7-2) and (4.1-3) will be used again when the rate of accumulation is not zero and unsteady-state heat transfer occurs. The mechanistic expression for Fourier s law in the form of a partial differential equation will be used where temperature at various points and the rate of heat transfer change with time. In Section 5.6 a general differential equation of energy change will be derived and integrated for various specific cases to determine the temperature profile and heat flux. 

Using the differential equation of energy change, derive the partial differential equation and boundary conditions needed for the case of laminar flow of a constant density fluid in a horizontal tube which is being heated. The fluid is flowing at a constant velocity u,. At the wall of the pipe where the radius. r = Tq, the heat flux is constant at. The process is at steady state and it is assumed at z = 0 at the inlet that the velocity profile is established. Constant physical properties will be assumed. 

Heat Transfer in a Solid Using Equation of Energy Change. A solid of thickness L is at a uniform temperature of Tq K. Suddenly the front surface temperature of the solid at z = 0 m is raised to T at r = 0 and held there and at z = L at the rear to T2 and held constant. Heat transfer occurs only in the z direction. For constant physical properties and using the differential equation of energy change, do as follows. 

Because we have a mathematical model of energy changes, we have a simple mathematical answer to this question. G is a state variable, so dG is an exact differential ( C.2.1). This means that, among other things, we can write the total differential as in Equation (4.41), or in the total energy form as... 

Because the potential energy term involves both Xi and X2, this appears to be an inseparable Hamiltonian function. In fact, in terms of x and Xy the differential equations of motion are coupled. However, a certain change of coordinates, or more precisely a coordinate transformation, leads to a separable form. This coordinate transformation comes about... 

In evaluating and/or designing compressors the main quantities that need to be calculated are the outlet (discharge) gas temperature, and the energy required to drive the motor or other prime mover. The latter is then corrected for the various efficiencies in the system. The differential equations for changes of state of any fluid in terms of the common independent variable are derived from the first two laws of thermodynamics ... 

An infinitesimal change in internal energy is an exact differential and is a unique function of temperature and pressure (for a given composition). Since the density of a given material is also uniquely determined by temperature and pressure (e.g., by an equation of state for the material), the internal energy may be expressed as a function of any two of the three terms T, P, or p (or v = 1/p). Hence, we may write ... 

The basic principle used to calculate the temperature in a compartment fire is the conservation of mass and energy. Since the energy release rate and the compartment temperature change with time, the application of the conservation laws will lead to a series of differential equations. 

The states of a dynamic system are simply the variables that appear in the time differential. For example, if we have a chemical reactor in which the concentration of reactant Ca and the temperature T change with time, the material balance for component A and the energy balance would give two differential equations ... 

Equation (7.85) frequently is called the Gibbs-Helmholtz equation. From it, the temperature coefficient of the free energy change (0AGm /. 7-/0T)p can be obtained if AGiji and AH are known. By differentiating Equation (7.83), we obtain... 

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