Tuesday, February 1, 2011



One of the fundamental laws of physics states that mass can neither be produced nor destroyed that is mass is conserved. Equally fundamental is the law of conservation of energy. Although energy in form, it can not be created or destroyed. These two laws of physics provide the basis for 2 tools which are used routinely in environmental engineering and science the mass balance and energy balances. This portion of the course deals with these tools mass balances are developed in some details in the following section, after which the concept of the energy balance is presented and applied.


This principle of conservation of mass is extremely useful. It means that if the amount of a pollutant some where (say, in a lake) increase, then that increase cannot be the result of some “ Magical” formation out of number where. The pollutant must have been either carried in to the lake from else where or produced via chemical reaction produced the mass increase in our pollutant, they must also have caused a corresponding decreasing in the mass of some other compound. Thus, conservation of mass allows us to compile a budget of the mass of our pollutant in the lake. This budget keeps track of the amounts of pollutants entering the lake, leaving the lake, and the amount formed or destroyed by chemical reaction. This budget can be balanced for a given time period. Similar to the way you might balance your check book.

(Mass at time t +Δt) = mass at time t) + (mass that entered from t to t + Δ t) – ( mass that existed from t to t+ Δt ) + ( net mass of pollutant produced from other compound by chemical reaction between t and t + Δ t)

Note that each term of this equation has unit of mass. This form of balance is most useful when there is clear and end to balance period , so that Δt is usually one month . In environmental work with values of mass flux—the rate at which mass enters or leaves a system. To develop an equation in terms of mass flux, the mass balance equation is Δt divided by Δt to produce an equation with units of mass per unit time. Dividing equation 12 by Δt and moving the first term an the right (mass at time ) to the left hand side yields the following equation.

(Mass at time t +Δt) -(mass at time t) = ( mass entering from t to t Δt) - (Mass exiting from t to t Δt) + (net chemical production between t and + Δt)

Note : Each term in this equation has units of mass/time. The left hand side of equation 2 is equal to Δm/ Δt). In the limit as Δt 0, this become dm/dt the rate of change of pollutant mass in the lake. We will refer to dm/dt as the accumulation rate of pollutant.as Δt 0, the first two terms on the in to the lake & the mass flux out of the lake production or loss. To stree the fact that each term in the new equation refers too flux or rate, we will use the symbol m to refer to a mass flux with units of mass\time the equation for mass balance is then
(mass accumulation rate) =(mass flux in )-(mass flux out)+(net rate of chemical production)
Dm/dt=min - mout +mreaction
Equation is the governing equation for the mass balance we will work in this course
In the reminder of this section, we will exam in the importance of carefully defining the
region over the which the mass balance is applied & discuss the terms of equation3. we
will then present examples of main types of situations for which mass balance are useful

The control volume:
A mass balance is only meaningful in the terms of a specific region of a space, which
as boundaries across which the term min & mout are determined this region is control

Terms of mass balance equation for a CSTR:

A well mixed tank in an analogue for many control volumes used in environmental
engineering. For Example in our lake it might be reasonable to assume that pollutants
dumped into lake are rapidly mixed through out the entire lake. In environmental
engineering &chemical engineering the term continuously stirred is used for such system
an example of CSTR is shown in below fig. we will usea mass balance for control
volume which encloses the CSTR in fig as an example to describe the meaning of each
term in equation3

Mass accumulation rate:

It is defined by dm/dt or Δt. The total mass in CSTR can not usually be measured. For
example CSTR represented an entire lake, measuring a total pollutant mass would require
analyzing all of the water in lake using concentration units of (mass)/(volume), the total
pollutant mass in the tank is equal to c,v , where v is the volume of CSTR. Thus, the
accumulation rate is equal to

Δm/ Δt = Δ(cv)/ Δt = v Δc/ Δt = vdc/dt

In steady state situation is one in which things do not change with time the incoming
Concentration &flow rate are constant, the outgoing flow rate is also constant &therefore
the concentration in the control volume is constant in steady state situation then dm/dt =
0 where as in nonsteady state situation –dm/dt is not 0

Mass flux in:
The example in fig1 include one pipe entry in the CSTR we will again use concentration
measured mass/volume in units to calculate the flux entering the CSTR through the pipe
Often, we know the volumetric floe rate, q, of each input stream for example in fig the
pipe as flow rate Q in , with corresponding pollutant concentration c in . the mass flux is
Given by

min = Q in.. c in
If it is not immediately clear how known Q*C gives a mass flux consider the units of
Each term

m = q .c
Mass/time = volume/time .mass/volume

Mass flux out:
The mass flux out of CSTR is similarly equal to product of volumetric flow rate in exist
Pipe time the concentration in exist pipe. Since the CSTR is well mixed, the
Concentration in the liquid leaving the CSTR is equal to the concentration is equal to
Concentration inside the CSTR. If the conventional to refer to the concentration with in

mout = cCSTR . Q out = C.Qout

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