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PTT 108: Mass and Energy Balances
1
CHAPTER 11BALANCES ON
TRANSIENT PROCESSES
By : Ms. Nor Helya Iman Bt Kamaludin
Email: [email protected]
PTT 108: Mass and Energy Balances
2
INTRODUCTIONTransient (unsteady-state):
Conditions at which the value of any system variable changes with time.
• Process systems: Batch system (no input and output streams) –
always transient Semibatch system (has input stream but no
output stream or vice versa) – always transient Continuous system (have input and output streams) – always transient when they are start up, shut down or become transient at other times due to planned or unexpected changes in operating conditions)
PTT 108: Mass and Energy Balances
3
General Balance Equation
Accumulation = Input + Generation -
Output - Consumption
Two forms of general balance equation;1) Differential balances
Relate instantaneous rates of change at a moment in time
2) Integral balances Relate changes that occur over a finite time
period
PTT 108: Mass and Energy Balances
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Differential BalancesStep 1: Write general balance equation
Step 2: Let consider the following terms of species A
(kg/s) : inlet mass flow rate (kg/s) : outlet mass flow rate (kg/s) : rate of generation (kg/s) : rate of consumption
PTT 108: Mass and Energy Balances
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Differential Balances (cont’d)
Step 3: Write a balance on A for a period of time from t to t + ∆t (suppose ∆t is
small enough), we get
Step 4: Suppose the mass of A in the system changes by an amount ∆M (kg) during
the small time interval. ∆M is the
accumulation of A.
PTT 108: Mass and Energy Balances
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Differential Balances (cont’d)Step 5: Divide by ∆t and let ∆t approach 0.
The ratio ∆M/ ∆t becomes the derivative of the
M wrt t (dM/ dt). The differential balance
equation become:
where M is the amount of balanced quantity in
the system and four terms on the right side are
the rates that may vary with time.Step 6: Write the complete balance equation
in expression for M(t)
t = 0 (initial condition) , M = … or simply M(0) = …
PTT 108: Mass and Energy Balances
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Integral BalancesRemember differential balance equation:
Step 1: Write the equation in form of
Step 2: Integrate from an t0 to tf to form integral
balance equation:
where left side is the accumulation and right side is
the amount of balanced quantity in the system
respectively.
PTT 108: Mass and Energy Balances
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Integral Balances (cont’d)
For a closed (batch) system, , thus the integral balance equation becomes:
Initial input + generation = final output + consumption
PTT 108: Mass and Energy Balances
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Balances on Single Well-Mixed Process Units
Procedure for writing and solving a transient material balance equation:
1. Eliminate terms in the general balance equation that equal zero Exp: input and output for batch systems, generation
and consumption for balances on total mass and nonreactive species.
2. Write an expression for the total amount of the balanced species in the system. Exp: [V(m3)p(kg/m3) for total mass, V(m3)CA (mol A/m3)
or ntotal (mol)xA (mol A/mol) for species A]
Differentiate the expression wrt time to obtain the
accumulation term in the balance equation.
PTT 108: Mass and Energy Balances
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Balances on Single Well-Mixed Process Units (Cont’d)
3. Substitute system variables into the remaining terms (input, generation, output, consumption) in the balance equation. Make sure that all the terms have the same units (kg/s. lb-mole/h, etc).
4. If y(t) is the dependent variable to be determined (e.g., the mass of the system contents, the conc of the species A, the mole fraction of the methane), rewrite the equation to obtain the explicit expression for dy/dt.
Boundary condition:for specified time : t = 0can be expressed as : t = 0, y = y0
or simply : y(0) = y0
PTT 108: Mass and Energy Balances
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Balances on Single Well-Mixed Process Units (Cont’d)
5. Solve the equation by analytically numerically
6. Check the solution using:(a) Substitute t = 0 and verify that the known
initial condition [y(0) = y0] is obtained.
(b) Find the long-time asymptotic (steady-state) value of the dependent variable, solving the resulting algebraic equation and verify.
(c) Differentiate solution to obtain an expression dy/dt, substitute and verify.
7. Use your solution to generate a plot or table of y versus t
PTT 108: Mass and Energy Balances
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Energy Balances on Single-Phase Nonreactive Processes
Step 1: General energy balance form: accumulation = input – output
where:accumulation =
input =
output =
(1)
(2)
(3)
PTT 108: Mass and Energy Balances
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Energy Balances on Single-Phase Nonreactive Processes (cont’d)Step 2: Substitute (1), (2) and (3) into general energy
balance and then divide by ∆t and let ∆t approach to
0. We obtain the general differential energy
balance:
*If there are several input and output streams, a term below
must be included in Eq. (4)(5)
(4)
PTT 108: Mass and Energy Balances
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Energy Balances on Single-Phase Nonreactive Processes (cont’d)Some consideration made to the systems:
1. The system has at most of the single input and output stream respectively where each has same mass flow rate.
2. Kinetic and potential energy changes in the system and between the inlet and outlet streams are negligible.
(7)
(6)
PTT 108: Mass and Energy Balances
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Energy Balances on Single-Phase Nonreactive Processes (cont’d)
Under the 2nd condition, Eq. (4) simplifies to
3. The temperature and the composition of the system contents do not vary with position within the system (the system is perfectly mixed). Thus, the outlet stream and the system contents must be at the same temperature
(9)
(8)
PTT 108: Mass and Energy Balances
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Energy Balances on Single-Phase Nonreactive Processes (cont’d)4. Consider the following conditions:
No phase changes or chemical reactions take place
within the system and are independent of pressure mean and of the system contents (and
the inlet
and outlet streams) are independent of composition
and temperature unchanging with time if Tr is a ref temp at which and M = mass (or
number of moles) of the system contents,
PTT 108: Mass and Energy Balances
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Energy Balances on Single-Phase Nonreactive Processes (cont’d)
Then, we get
Finally substitute the expressions of (6) through (10) into (4) to obtain
(10)
PTT 108: Mass and Energy Balances
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Energy Balances on Single-Phase Nonreactive Processes (cont’d)General Energy Balance for Open
System:
General Energy Balance for Closed System: