This principle of conservation of mass is extremely useful. It means that if the amount of a pollutant somewhere (say, in a lake) increases, then that increase cannot be the result of some ``magical'' formation out of nowhere. The pollutant must have been either carried into the lake from elsewhere or produced via chemical reaction from other compounds that were already in the lake. And, if chemical reactions produced the mass increase in our pollutant, they must also have caused a corresponding decrease in the mass of some other compounds. 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 pollutant 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 checkbook:
Note that each term of this equation has units of mass. This form of balance is most useful when there is a clear beginning and end to the balance period, so that is meaningful. For example, in a checkbook balance is usually one month. In environmental problems, however, it is usually more convenient to 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 divided by to produce an equation with units of mass per unit time. Dividing equation 12 by and moving the first term on the right (mass at time t) to the left hand side yields the following equation.
Note that each term in this equation has units of mass/time. The left hand side of equation 13 is equal to . In the limit as , this becomes , the rate of change of pollutant mass in the lake. We will refer to as the accumulation rate of the pollutant. As , the first two terms on the right side of equation 13 become the mass flux into the lake and the mass flux out of the lake. The last term of equation 13 is the rate of chemical production or loss. To stress the fact that each term in the new equation refers to a flux or rate, we will use the symbol to refer to a mass flux with units of mass/time. The equation for mass balances is then
Equation 14 is the governing equation for the mass balances we will work with in this course.
In the remainder of this section, we will examine the importance of carefully defining the region over which the mass balance is applied and discuss the terms of equation 14. We will then present examples of the main types of situations for which mass balances are useful.
A mass balance is only meaningful in terms of a specific region of space, which has boundaries across which the terms and are determined. This region is called the control volume. In our derivation of the mass balance equation, we have referred to the mass of pollutant in a lake and the fluxes of pollutant into and out of the lake---that is, we have used a lake as our control volume. Theoretically, any volume of any shape and location can be used. Realistically, however, certain control volumes are more useful than others. The most important attribute of a control volume is that it have boundaries over which you can calculate and .
A well-mixed tank is an analogue for many control volumes used in environmental engineering. For example, in our lake example it might be reasonable to assume that pollutants dumped into the lake are rapidly mixed throughout the entire lake. In environmental engineering and chemical engineering, the term Continuously Stirred Tank Reactor, or CSTR is used for such a system. An example of a CSTR is shown in Figure 1. We will use a mass balance for a control volume which encloses the CSTR in Figure 1 as an example to describe the meaning of each term in equation 14.
The mass accumulation rate is, by definition, , or . The total mass in the CSTR cannot usually be measured. For example, if the CSTR represented an entire lake, measuring the total pollutant mass would require analyzing all of the water in the lake. However, our assumption that the CSTR is well-mixed means that this is not necessary. If the tank is well-mixed, then the concentration of our pollutant is the same everywhere in the tank, and we need only to measure the concentration in a sampling from the tank. Using concentration units of (mass)/(volume), the total pollutant mass in the tank is equal to , where V is the volume of the CSTR. Thus, the accumulation rate is equal to
Here, we have made the assumption that the volume of the CSTR is constant. This is usually a reasonable assumption for liquids, although it may not always be valid for gases. However, will always be equal to .
Mass balance problems can be divided into those that are in steady state and those that are non-steady state. A steady-state situation is one in which things do not change with time---the incoming concentration and flow rate are constant, the outgoing flow rate is constant, and therefore the concentration in the control volume is constant. For steady-state systems, then, . Non-steady-state conditions result whenever flows start or stop, or when the concentration in an incoming stream changes. For non-steady-state situations, is nonzero.
The example in Figure 1 includes one pipe entering the CSTR. We will again use concentration measured in mass/volume units to calculate the flux entering the CSTR through the pipe.
Often, we know the volumetric flow rate, Q, of each input stream. For the example of Figure 1 the pipe has a flow rate of , with corresponding pollutant concentration of . The mass flux is then given by
If it is not immediately clear how one knows that gives a mass flux, consider the units of each term:
If the volumetric flow rate is not known, it may be calculated from other parameters. For example, if the fluid velocity v and the cross-sectional area A of the pipe are known, then .
Another way to describe the flux is in terms of a flux density J times the area through which the flux occurs. J has units of , and we will study it in more detail when we cover diffusion in section 3. This type of flux notation is most useful at interfaces where there is no fluid flow, such as the interface between the air and water and the surface of a lake.
The flux out of the CSTR is similarly equal to the product of volumetric flow rate in the exit pipe times the concentration in the exit pipe. Since the CSTR is well-mixed, the concentration in the liquid leaving the CSTR is equal to the concentration inside the CSTR. It is conventional to refer to the concentration within the CSTR simply as C. Thus,
The term refers to the net rate of production of our pollutant from chemical reactions, in units of mass/time. Thus, if other compounds react to form our pollutant, will be greater than zero; if our pollutant reacts to form some other compounds, resulting in a loss of the pollutant, will be negative. Production or loss of a compound by a chemical reaction is usually described in terms of concentration, not mass. So it is necessary to multiply the chemical rate of change of concentration by the volume of the CSTR to obtain units of mass/time:
There are a number of possibilities for the form of , and the resulting . The most common include:
Reactor Analysis refers to the use of mass balances to analyze pollutant concentrations in a control volume which is a chemical reactor. Do not let the term ``reactor'' fool you, however. The reactor can be any control volume we want it to be. So the term reactor analysis is used to describe the application of the mass balance process to environmental situations also. Reactor analyses can be divided into two types: CSTRs (Continuously Stirred Tank Reactors) and PFRs, or Plug Flow Reactors. We have defined CSTRs already---they are simply well-mixed tanks which are used to model well-mixed environmental reservoirs. Plug Flow Reactors are essentially pipes, and they are used to model things like rivers, in which fluid is not mixed in the upstream-downstream direction.
In this section, we will present examples of the types of situations CSTRs are used to model. Plug Flow Reactors are described and used in examples in the following section. Example 2.1 demonstrates the use of CSTR analysis to determine the concentration of a substance resulting from the mixing of two or more influent flows. This type of calculation will be used again in the third part of this course to determine the initial BOD loading in a river downstream of a sewage outflow. Examples 2.2 through 2.4 refer to the tank in Figure 1 and demonstrate steady-state and non-steady-state situations with and without first-order chemical decay. Calculations completely analogous to those in examples 2.2, 2.3, and 2.4 can be used to determine the concentration of sewage pollutants exiting a treatment reactor, the rate of increase of pollutant concentrations within a lake resulting from a new pollutant source, and the period required for pollutant levels to decay from a lake or reactor once the source is removed.
Before beginning our analysis, we should determine whether this is a steady-state or non-steady-state problem, and whether the chemical reaction term will be nonzero. Since the problem statement does not refer to time at all, and it seems reasonable to assume that both the river and sewage have been flowing for some time and will continue to flow, this is a steady-state problem. Sewage does participate in chemical and biological reactions. However, we are interested here in mixing---that is, in what concentration results right after the two flows mix. So we will assume that the mixing occurs instantly, without sufficient time for any reactions to occur.
where the term has been set to zero because we are ignoring chemical reaction. Since this is a steady-state problem, . Therefore, as long as the density is constant, , or .
Plugging in, we find that
Solving for C, we find that
The numerical solution is
Because of the extra term on the right (), this equation cannot be immediately solved in the way that example 2.4 was solved. However, if we make a change of variables, we can make the form of this equation similar to that of example 2.4. Let . Since is constant, . Therefore, the last equation above is equivalent to
Rearranging and integrating,
which yields
If we now substitute for y, we obtain
The second equation is obtained using the observation that , since the tank is started clean. Rearranging, we can obtain
This is the solution to the question posed in the problem statement.
Note what happens as : \
and . This is not surprising, since this is a
conservative substance. If we run the reactor for a long enough
period, the concentration in the reactor will eventually reach the
inlet concentration. Using the equation we have derived for C as a
function of time, we could determine how long it would take for
the concentration to reach, say, 90% of the inlet value.
which yields
Since is equal to ,
and, exponentiating both sides,
Plugging in the values from the problem, with equal to the steady-state solution of 32 mg/l yields
Taking the natural logarithm of both sides,
The Plug Flow Reactor (PFR) is used to model the chemical transformation of compounds as they are transported in ``pipes.'' The ``pipe'' may represent a river, a region between two mountain ranges through which air flows, or a variety of other conduits through which liquids or gases flow. Of course, it can even represent a pipe. A schematic diagram of a PFR is shown in Figure 3.
As fluid flows down the PFR, the fluid is mixed in the radial direction, but mixing does not occur in the axial direction---each plug of fluid is considered a separate entity as it flows down the pipe. However, as the plug of fluid flows downstream, time passes. Therefore, there is an implicit time dependence even in steady-state PFR problems. However, because the velocity of the fluid in the PFR is constant, time and downstream distance are interchangeable: . We will use this observation together with the mass balance formulations we have worked with already to determine how pollutant concentrations vary during flow down a PFR.
To develop the equations which describe pollutant concentration in the plug of fluid as it flows down the PFR, we will conduct a mass balance on a control volume which encloses a section of the PFR of infinitesimally small thickness dx, as shown in Figure 4. Since the thickness is small, we can assume that the fluid in that region of the PFR is well-mixed. The mass balance equation for this control volume is
We have set equal to zero, indicating that this is a steady-state problem. We are assuming here that conditions at a given location in the PFR are constant. Concentrations can still vary along the PFR, however.
Noting that the volume of our control volume is given by , dividing by dx, and rearranging, we obtain
In the limit as , the left hand side becomes the derivative , so we obtain
As discussed earlier, can take a variety of forms, depending on the type(s) of chemical reaction that are occurring.
which can be integrated as follows:
Therefore, for a PFR of length l,
where the volume of the PFR, V, is equal to the length times time area.
Equation 26 describes the way in which concentration decreases during passage down a PFR with loss via a first-order reaction. Note that, since the time which passes during transport down the PFR is equal to , equation 26 is equivalent to
which is the solution to the differential equation which describes the loss of a pollutant by first-order kinetics: . That is, in a plug flow reactor time and distance are interchangeable, and the concentration at any location in the PFR may be calculated simply by determining the chemical decay during the time it took to reach that location.
The CSTR and the PFR are fundamentally different. When a parcel of fluid enters the CSTR, it is immediately mixed throughout the entire volume of the CSTR. In contrast, each parcel of fluid entering the PFR remains separate during its passage through the reactor. This difference results in differing behavior. We will look at these differences for one special case: the continuous addition of a pollutant to each reactor, with destruction of the pollutant within the reactor according to first-order kinetics. The two reactors are shown in Figure 5.
We will assume that the incoming concentration (), the flow rate (Q), and the first-order reaction rate constant (k) are given and are the same for both reactors. Then, we will consider two common problems: (1) if we know the volume V (same for both reactors), what is the resulting outlet concentration ()? and (2) if we need a specified outlet concentration, what volume of reactor is required? Table 2 summarizes the results of this comparison.
The results shown in Table 2 indicate that, for equal reactor volumes, the plug flow reactor is more efficient that the CSTR and, for equal outlet concentrations, a smaller PFR is required. Why is this? The answer has to do with the fundamental difference between the two reactors. In a PFR, each and every molecule spends the same amount of time in the reactor; that period is equal to . Since first-order decay occurs according to , the concentration in each parcel of fluid entering the reactor drops by this amount. In contrast, in a CSTR there is no single amount of time that each small parcel of fluid spends in the reactor. Some parcels may spend a long time mixing around inside the CSTR; other parcels may, by chance, reach the exit in a relatively short time. Since all these parcels are mixed together and result in a single outlet concentration, an average value of \ results.
To see why that average value is higher than the corresponding value for a PFR, consider what happens when is equal to 2, approximately the value in the first example of Table 2. Then, . This is the value of \ that would result in the PFR. Let's assume that we can model the mixing in the CSTR by splitting the fluid entering the CSTR into two parcels. The first parcel remains in the CSTR only one quarter of the time a parcel would take to pass through the PFR, while the second parcel remains in the CSTR four times as long as it would in the PFR. (So the average time spent in the CSTR by the two parcels is the same as the time spent in the PFR---both are equal to .) The concentration in the first parcel when it reaches the CSTR exit is determined by its value of kt, which is 4 times larger than the value for the PFR: . The concentration in the second parcel is reduced less, because it spends a shorter time in the reactor: . The actual concentration in the exit of the CSTR in this situation would be the average of the concentrations in the two parcels, so .
Thus, the resulting value of for the CSTR is higher than that for the PFR (0.30 versus 0.14), even though the average residence time is the same for both reactors. The reason for this is illustrated in Figure 6, and results from the fact that concentration decays exponentially with time for a first-order reaction. Thus, the parcel that spends a shorter period of time in the CSTR exits with a concentration that is increased significantly relative to the PFR. However, the parcel that spends a longer period in the CSTR exits with a concentration that is decreased only a small amount (again, relative to the PFR).
Solving for V, we obtain
As expected, this volume is smaller than the 500 m required for
the CSTR in example 2.2.
A number of terms are used to describe the average period spent in a given reactor. The terms retention time, detention time, and residence time are all used to refer to , the average period spent in the reactor. This parameter has units of time. As discussed above, for a plug flow reactor the retention time is actually the time spent in the reactor. However, for a CSTR the retention time is the average period spent in the reactor.
The reciprocal of the retention time, , has units of inverse time---the same units as a first-order rate constant. This value is sometimes referred to as the exchange rate.
For the PFR in example 2.5,
Modern society is dependent on the use of energy. Such use requires transformations in the form of energy and control of energy flows. For example, when coal is burned at a power plant, the chemical energy present in the coal is converted to heat, which is then converted in the plant's generators to electrical energy. Eventually, the electrical energy is converted back into heat for warmth or used to turn motors. However, energy flows and transformation are also the cause of environmental problems. Thermal heat energy from electrical power plants can result in increased temperature in rivers used for cooling water; ``greenhouse'' pollutants in the atmosphere alter the energy balance of the earth and may cause significant increases in global temperatures in the future; and many of our uses of energy are themselves associated with emissions of pollutants.
We can keep track of the movement of energy and changes in its form using energy balances, which are analogous to the mass balances we discussed in the previous section. We can do this because of the law of conservation of energy which states that energy can neither be produced nor destroyed. (Conservation of energy is sometimes referred to as the first law of thermodynamics.) As long as we consider all the possible forms of energy, there is no term in energy balances which is analogous to the chemical reaction term in mass balances. That is, we can treat energy as a conservative substance.
The forms of energy can be divided two types: internal energy and external energy. Energy which is a part of the molecular structure or organization of a given substance is internal. Energy which results from the location or motion of the substance is external. Examples of external energy include gravitational potential energy and kinetic energy. Gravitational potential energy is the energy gained when a mass is moved to a higher location above the earth. Kinetic energy is the energy which results from the movement of objects. When a rock thrown off of a cliff accelerates toward the ground, the sum of kinetic and potential energy is conserved (neglecting friction)---as it falls it loses potential energy, but increases in speed, gaining kinetic energy. Examples of some common forms of energy are given in Table 4.
Heat is a form of internal energy. It results from the random motions of atoms. Heat is thus really a form of kinetic energy, although it is considered separately. When you heat a pot of water, you are adding energy to the water. That energy is stored in the form of internal energy, and the change in internal energy of the water is given by
where C is the heat capacity or specific heat of the water, with units of [energy]/([mass][temperature]). Heat capacity is a property of a given material. For water, the heat capacity is 1 BTU/(), or 4184 J/().
Chemical internal energy reflects the energy in the chemical bonds of a substance. This form of energy is composed of two parts:
In analogy with the mass balance equation (equation 14), we will use the following equation to conduct energy balances:
We will illustrate the use of this relationship with some examples.
Each term of this equation is an energy flux, and has the units of (energy/time). To solve, we need to use the same units in each term. We will use the definition of watts: watts are defined as Joules/s. In addition, we need to convert the water flow rate (gallons/min) to mass of water per unit time, using the density of water. Combining the first and third terms we obtain
which is a cold shower! (You may have foreseen this answer if you
have ever taken a shower after the hot water in the tank was used up
by previous showerers.)
We will solve this for , given that is equal to C.
The two previous examples related to the controlled conversion and transfer of energy for a beneficial use. However, the use of energy for heat always results in some loss to the environment due to imperfect insulation, resulting in higher energy use or less heating than one would calculate. In addition, the second law of thermodynamics states that it is impossible to convert heat energy to work with 100% efficiency. Conversion of heat to work is essentially what is done in the generator of an electric power plant, and as a result a significant fraction of the energy released from fuel combustion is lost during the conversion. Modern large power plants convert fuel energy to electricity with an overall efficiency in the 30--35% range. The next example looks at what happens to the heat energy that is not converted to electricity. Finally, example 2.11 considers the implications of another aspect of the burning of fossil fuels in power plants, vehicles, and for heating. Whenever fossil fuels are burned, carbon atoms in the fuel are converted to carbon dioxide () and released into the atmosphere. As a result of this process, the concentration in the atmosphere is increasing at a rate of about 1 ppmv/year. Carbon dioxide contributes to the greenhouse effect, which is considered in example 2.11.
Rearranging, we obtain
The remainder of this problem is basically a problem of unit conversions. To obtain requires multiplication of the given river volumetric flow rate by the density of water (1000 kg/m). We also use the heat capacity of water, C. Thus,
The energy flux in is equal to the solar energy intercepted by the earth. At the earth's distance from the sun, the sun radiates 342 W/m. We will refer to this value as S. The earth intercepts an amount of energy equal to S times the cross-sectional area of the earth: . However, because the earth reflects part of this energy back to space, is equal to only 70% of this value:
The second term, , is equal to the energy radiated to space by the earth. The energy emitted per unit surface area of the earth is given by Boltzmann's Law:
where is Boltzmann's constant, equal to . To obtain , we multiply this value by the total surface area of the earth, . (We use the total surface area of the sphere here because energy is radiated away from the earth during both day and night.)
We can now solve our energy balance by setting equal to .
Plugging in the values for S and and taking the fourth root yields an average temperature of T=255 K, or -18 C.
This is too cold! In fact, the globally averaged temperature at the surface of the earth is much warmer: 287 K. The reason for the difference is the presence of gases in the atmosphere that absorb the infrared radiation emitted by the earth and prevent it from reaching space. We neglected these gases in our energy balance. However, if we denote the energy flux absorbed and retained by these gases by , we can then correct our value for :
The reduction in which results from greenhouse gas
absorption is sufficient to cause the higher observed surface
temperature. Clearly, this is largely a natural phenomenon---surface
temperatures were well above 255 K long before people began
burning fossil fuels. The main natural greenhouse gas is water vapor.
However, increasing atmospheric concentrations of carbon dioxide and
other gases emitted by human activities are increasing the value of
. So far, this increase is approximately 2 W/m\
averaged over the entire earth, and projections indicate that the
increase could be as high as 5 W/m over the next 50 years.
According to our energy balance, this increase in is
expected to result in an increase in the globally averaged
temperature. (There is considerable uncertainty in the precise value
of the resulting increase, however, due to a number of complexities
that we have not considered.)
.. A pond is used to treat sewage wastewater before the liquid is discharged into a river. The inflow to the pond is sewage at a flow rate of and with a BOD concentration of . The volume of the pond is 20,000 m. The purpose of the pond is to allow time for the decay of BOD to occur before discharge into the environment. BOD decays in the pond with a first-order rate constant equal to 0.25/day. What is the BOD concentration at the outflow of the pond, in units of mg/l?
answer: 11 mg/l
.. For each of the following problems, would you use a steady-state or non-steady-state mass balance to obtain a solution? For each situation, also indicate whether the compound for which you would conduct a mass balance is conservative or non-conservative. Give an explanation for each of your answers. (You do not need to actually solve these problems.)
answer: (a) non-steady state
.. A mixture of two gas flows is used to calibrate an air pollution measurement instrument. The calibration system is shown in Figure 7. If the calibration gas concentration is 4.90 ppmv, the calibration gas flow rate is 0.010 l/min, and the total gas flow rate is 1.000 l/min, what is the concentration of calibration gas after mixing ()? (Assume that the concentration upstream of the mixing point is zero.)
answer: 49.0 ppbv
.. You are in an old spy movie, and have been locked into a small room (volume 1000 ft). You suddenly realize that a poison gas has just started entering the room through a ventilation duct. Recognizing the type of poison from its smell, you know that if the gas reaches a concentration of 100 mg/m, you will die instantly, but that you are safe as long as the concentration is less than 100 mg/m. If the ventilation air flow rate in the room is 100 ft/min and the incoming gas concentration is 200 mg/m, how long do you have to escape?
answer: 6.9 minutes
..
Sewage waste is added to a stream
through a discharge pipe. The river flow rate upstream of the
discharge point is . The discharge occurs at a
flow of and has a BOD concentration of 50.0 mg/l.
Assuming that the upstream BOD concentration is negligible
(a) What is the BOD concentration just downstream of the discharge point?
(b) If the stream has a cross sectional area of 10 m, what would
the BOD concentration be 50 km downstream? (BOD is removed with a
first-order decay rate constant equal to 0.20 day)
answer: (a) 4.7 mg/l. (b) 4.2 mg/l
..
(a) Calculate the hydraulic residence times (the retention time) for Lake
Superior and for Lake Erie using the data in
Table 3.
(b) Assume that both lakes currently are polluted with the same
compound at a concentration which is 10 times the maximum acceptable
level. If all sources of the compound are removed, how long will it
take the concentration to reach acceptable levels in each lake?
Assume that the pollutant does not decay chemically.
(c) Comment on the significance of your answers.
answer: (a) Lake Superior: 179 years; Lake Erie: 3 years. (b) Lake Superior: 412 years; Lake Erie: 6 years.
.. How many watts of power would it take to heat 1 liter of water (weighing 1.0 kg) by 10 C in 1.0 hour? Assume that no heat losses occur, so that all of the energy expended goes into heating the water.
answer: 12 Watts