The General Balance Equation... Again

Two forms of this equation were discussed differential balances, which relate instantaneous rates of change at a moment in time, and integral balances, which relate changes that occur over a finite time period. We examine in this section the nature of the relationship between these two types of balances in doing so, we belatedly show why they are called differential and integral. 

Suppose a species A is involved in a process. Let mi (kg/s) and moui(kg/s) be the rates at which A enters and leaves the process by crossing the boundaries, and let rgen(kg/s) and rconsfkg/s) be the rates of generation and consumption of A within the system by chemical reaction. Any or all of the variables mout gen, and rcons niay vary with time. 

Let us now write a balance on A for a period of time from r to f + At, supposing that At is small enough for the quantities mjn, mout fgen nd /cons to be considered constant. (Since we will eventually let At approach 0, this assumption is not restrictive.) The terms of a balance on A are easily calculated. 

We also suppose that the mass of A in the system changes by an amount AW(kg) during this small time interval. By definition, AAf is the accumulation of A in the system. From the balance 

If now we divide by At and then let At approach 0, the ratio AAf/At becomes the derivative of M with respect to t (dM/ dt), and the balance equation becomes 

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