Mass balance equation left-hand side

The Navier-Stokes equations have a complex form due to the necessity of treating many of the terms as vector quantities. To understand these equations, however, one need only recognize that they are not mass balances but an elaboration of Newton s second law of motion for a flowing fluid. Recall that Newton s second law states that the vector sum of all the forces acting on an object ( F) will be equal to the product of the object s mass (m) and its acceleration (a), or XF = ma. Now consider the first of the three Navier-Stokes equations listed above, Eq. (10). The object in this case is a differential fluid element, that is, a small cube of fluid with volume dx dy dz and mass p(dx dy dz). The left-hand side of the equation is essentially the product of mass and acceleration for this fluid element (ma), while the right-hand side represents the sum of the forces... 

Here, the enthalpy of the products of mass flowrate G and specific heat c is measured relative to T0, the inlet temperature of the reactants. The term for rate of heat generation on the left-hand side of this equation varies with the temperature of operation T, as shown in diagram (a) of Fig. 1.20 as T increases, lA increases rapidly at first but then tends to an upper limit as the reactant concentration in the tank approaches zero, corresponding to almost complete conversion. On the other hand, the rate of heat removal by both product outflow and heat transfer is virtually linear, as shown in diagram (b). To satisfy the heat balance equation above, the point representing the actual operating temperature must lie on both the rate of heat production curve and the rate of heat removal line, i.e. at the point of intersection as shown in (c). 

This value can now be plugged back into the first mass balance equation, which we previously rearranged with d on the left hand side, Equation (12) ... 

In chemical reactions, there is always a mass balance, i.e., number of atoms of any species on the left-hand side of an equation is the same as that of the species on the right-hand side. In an electrochemical reaction, in addition to the mass balance, there is also a charge balance in which the total charge on the left-hand side of an equation is the same as that on the right-hand side. Thus, mass and charge balance are two basic requirements of an electrochemical equilibrium. 

To close the present derivation of the continuum population balance equations, one needs to simplify the last two terms on the left-hand side of Equations (A-23) and (A-24). These terms describe various mechanisms of mass and/or bubble transfer among the regions defined by the characteristic functions (A-2)-(A-4). 

This is the relation for the momentum balance in the x-direction, and is known as the x-momentnm equation. Note that we would obtain the. same result if we used momentum flow rates for the left-hand side of this equation instead of mass limes acceleration. If there is a body force acting in the x-direction, it can be added to the right side of the equation provided that it is expressed per unit volume of the fluid. 

When we write down a chemical reaction in the form of an equation, we have to balance the equation to ensure that we don t make mass appear or disappear. That is to say, the number of atoms of any particular element that are present at the beginning of the reaction, on the left-hand side or LHS of the arrow, is equal to the number on the right-hand side (RHS). Looking at a simple equation such as the combustion or burning of hydrogen ... 

Where t is time, z are the axial position in the column, qt is the concentration of solute i in the stationary phase in equilibrium with Cu the mobile phase concentration of solute /, u is the mobile phase velocity, Da is the apparent dispersion coefficient, and F is the phase ratio (Vs/Vm). The equation describes that the difference between the amounts of component / that enters a slice of the column and the amount of the same component that leaves it is equal to the amount accumulated in the slice. The fist two terms on the left-hand side of Eq. 10 are the accumulation terms in the mobile and stationary phase, respectively [109], The third term is the convective term and the term on the right-hand side of Eq. 10 is the diffusion term. For a multi component system there are as many mass balance equation, as there are active components in the system [13],... 

Figure 2-2. An arbitrarily chosen control volume of fixed position and shape immersed in a fluid with velocity u. The velocity of the surface of the control volume is zero, and thus there is a net flux of fluid through its surface. Equation (2-3) represents a mass balance on the volume V, with the left-hand side giving the rate of mass accumulation and the right-hand side the net flux of mass into V that is due to the motion u.

Answer (a) The mass number and atomic number are 212 and 84, respectively, on the left-hand side and 208 and 82, respectively, on the right-hand side. Thus, X must have a mass number of 4 and an atomic number of 2, which means that it is an a particle. The balanced equation is... 

Geometry and the Left-Hand Side of the Mass Balance Equation... 

If the contactor is at steady state, the left-hand side of each equation is identically zero. Adding the two equations and placing the terms for the heavy phase and the light solvent phase on opposite sides of the equation lead to the "steady-state" material balance we had before Now we can see where the rates of mass transfer come in. 

When all terms are grouped on the left-hand side of equation (1-10), the rearranged mass balance for component i. 

The second and fourth terms on the left-hand side of (9-15) can be combined to reveal the equation of continuity (i.e., the overall mass balance) for the mixture ... 

The only assumption is that the physical properties of the fluid (i.e p and A.mix) are constant. The left-hand side of equation (11-1) represents convective mass transfer in three coordinate directions, and diffusion is accounted for via three terms on the right side. If the mass balance is written in dimensionless form, then the mass transfer Peclet number appears as a coefficient on the left-hand side. Basic information for dimensional molar density Ca will be developed before dimensionless quantities are introduced. In spherical coordinates, the concentration profile CA(r,6,4>) must satisfy the following partial differential equation (PDE) ... 

Previously, the mass balance was written for reactant A via equation (19-9), whose stoichiometric coefficient va is -1. Equation (19-16) is rearranged such that all species-specific quantities are grouped together on the left-hand side of the following equation ... 

For a nuclear reaction to be balanced, the sum of all the atomic numbers on the left-hand side of the reaction arrow must equal the sum of all the atomic numbers on the right-hand side of the arrow. The same is true for the sums of the mass numbers. Here s an example Suppose you re a scientist performing a nuclear reaction by bombarding a particular isotope of chlorine (Cl-35) with a neutron. (Work with me here. I m just trying to get to a point.) You observe that an isotope of hydrogen, H-1, is created along with another isotope, and you want to figure out what the other isotope is. The equation for this example is... 

Now knowing the solution for the overall mass balance, we seek a solution for the component balance. It can also be rearranged for solution by separation of variables. Expanding the left-hand side of equation (1.3.8) gives... 

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