Assignment 1 - 2013
Due date(s): | 21 January 2013 |
(PDF) | Assignment questions |
(PDF) | Assignment solutions |
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- Assignment objectives**: math refresher; chemistry refresher; review mol balances
* Always state assumptions in this assignment, midterms and exams. * Never use an equation by just writing it down; state its origin and all simplifying assumptions. *For example*: using the general mol balance in a batch reactor, under the assumption of a well-mixed and constant volume system, we have: :math:`\dfrac{dN_j}{dt} = r_jV`
.. question:: :grading: 10
#. :math:`\displaystyle \int{ \frac{1}{x} \,dx} =` #. :math:`\displaystyle \int{ \frac{1}{x^2} \,dx} =` #. :math:`\displaystyle \int{ \frac{1}{ax+b} \,dx} =` #. :math:`\displaystyle \int{ \frac{1}{\sqrt{x}} \,dx} =` #. When do we require an integration constant; and when do we not require it?
.. answer::
#. :math:`\displaystyle \int{ \frac{1}{x} \,dx} = \ln{x} + C` #. :math:`\displaystyle \int{ \frac{1}{x^2} \,dx} = -\frac{1}{x} + C` #. :math:`\displaystyle \int{ \frac{1}{ax+b} \,dx} = \frac{1}{a}\ln(ax+b) + C` #. :math:`\displaystyle \int{ \frac{1}{\sqrt{x}} \,dx} = 2\sqrt{x} + C` #. We require an integration constant when the limits of integration are unspecified (indefinite integrals); otherwise we don't require the constant, since the integral is uniquely defined.
.. question:: :grading: 10
#. A vessel contains a gas of concentration :math:`20\,\text{mol.m}^{-3}`. The gas is stored at 375°C. Assuming this is an ideal gas, what is the pressure in the vessel measured in kPa? What assumption are you making (apart from the ideal-gas law)?
#. A constant volume batch reactor operates at 14.7 psi and 1340°F. The reactor volume is :math:`290\,\text{ft}^3`. How many mols are in the system, assuming an ideal gas?
.. answer::
Assumptions: Pure species in tank, at constant volume and temperature.
#. :math:`T = 648\,\text{K}`, so from the ideal gas law: :math:`P = CRT = (20\,\text{mol.m}^{-3})(8.314\,\text{J.mol}^{-1}.K^{-1})(648\,\text{K}) = 107.7\,\text{kPa}`.
#. At :math:`P = 14.7\,\text{psi} = 1\,\text{atm}`, with :math:`V = 290\,\text{ft}^3 = 8.205\,\text{m}^3` and :math:`T = 1000\,\text{K}`, then :math:`R = 8.205746 \times 10^{-5}\, \text{m}^3\text{.atm.K}^{-1}\text{.mol}^{-1}`, giving :math:`n = 100\,\text{mol}`.
.. question::
:grading: 10
Milk is pasteurized if it is heated to 63°C for 30 min, but if it is heated to 74°C it only needs 15 seconds for the same result. Find the activation energy of this sterilization process.
Recall the activation energy for a chemical reaction is the :math:`E` term, and the rate constant in is given by :math:`k = k_0 e^{\frac{-E}{RT}}`.
*Hint*: assume pasteurization proceeds via first-order kinetics; what is the "reactant"?
.. answer::
To ask for the activation energy of a process means assuming an Arrhenius temperature dependency for the process. Here we are told that
* :math:`t_1 = 1800` seconds are required at temperature :math:`T_1` = 336 K * :math:`t_2 = 15` seconds are required at temperature :math:`T_2` = 347 K
As told, assuming first order kinetics, :math:`-r_A = k_A C_A = k_0 e^{\frac{-E}{RT}} C_A`.
For a batch system, making the regular assumption of constant volume and well-mixed (both are suitable for pasteurization):
.. math::
t = \int_{C_{A0}}^{C_A}{\frac{dC_A}{-r_A}} = \int_{C_{A0}}^{C_A}{\frac{dC_A}{-k_A C_A}} = \frac{1}{k_A}\ln\left(\frac{C_A}{C_{A0}} \right)
So for the 2 systems we have that :math:`C_{A0}` and :math:`C_{A}` are the same, so we can write the ratio:
.. math::
\dfrac{t_1}{t_2} = \frac{1800}{15} = 120 = \frac{k_{A,2}}{k_{A,1}} = \frac{k_{0} e^{\frac{-E}{RT_2}}}{k_{0} e^{\frac{-E}{RT_1}}}
Simplifying, recognizing that :math:`k_0`, and :math:`R` are the same for both systems:
.. math::
120 &= \exp\left(-\frac{E}{RT_2} + \frac{E}{RT_1} \right) \\ \ln(120) &= -\frac{E}{R}\left(\frac{1}{347} - \frac{1}{336}\right)\\ -E &= \frac{\ln(120)(8.314)}{-9.43\times 10^{-5}}\\ E &= 421 885\,\text{J.mol}^{-1}
.. question::
:grading: 13
The fermentation of an active ingredient :math:`A` is to be carried out in a reactor. The reaction kinetics are given by:
.. math::
A \longrightarrow R
-r_A = \frac{0.1 C_A}{1+0.5 C_A } \left[\dfrac{\text{mol}}{\text{L.min}}\right]
1. Consider a batch reactor filled with 750 L of reactant at :math:`C_{A,0} = 2\,\text{mol.L}^{-1}`. How long must the reactor be operated to achieve an exit concentration of A of :math:`0.1\,\text{mol.L}^{-1}`?
If the feed rate is continuously fed at :math:`25\,\text{L.min}^{-1}`, with :math:`C_{A,0} = 2\,\text{mol.L}^{-1}`. Determine the volume required for a
2. CSTR 3. PFR
to achieve an exit concentration of A of :math:`0.1\,\text{mol.L}^{-1}`.
4. Which of the CSTR or PFR require a smaller volume?
.. answer::
#. For a well-mixed batch reactor, the design equation is:
.. math:: \dfrac{dC_A}{dt} &= r_A \\ -\int_{C_{A0}}^{C_A}{\dfrac{1+0.5C_A}{0.1C_A}dC_A} &= \int_{t=0}^{t}{dt}\\ -10\ln\left(\dfrac{0.1}{2.0}\right) - 5.0\left(0.1 - 2.0\right) &= t\\ t &= 29.95 + 9.5 = 39.45\,\text{minutes}
#. Using the design equation for CSTRs (assuming it to be well-mixed and operating at steady state, with a constant volumetric flow rate of :math:`25\,\text{L.min}^{-1}`):
.. math::
V &= \dfrac{F_{A0}-F_A}{-r_A} \\ &= \dfrac{50-2.5}{\frac{(0.1)(0.1)}{1+(0.5)(0.1) }} \\ &= 4987.5 \approx 5000\,\text{m}^3
#. Use the mole balance equation for PFRs, making the usual assumption of steady state, and well-mixed in the radial direction, and using that :math:`(q)(dC_A) = dF_A`, assuming constant volumetric flow throughout the PFR:
.. math::
\dfrac{dF_A}{dV} &= r_A \\ dV &= q\dfrac{dC_A}{r_A} \\ V &= q \int_{C_{A0}}^{C_A}{-\frac{1.0 + 0.5C_A}{0.1C_A}dC_A}\\ V &= -(25)(10) \ln\left(\dfrac{0.1}{2.0}\right) -(25)(5)(0.1 - 2.0)\\ V &= 986\,\text{L}
.. question:: :grading: 7
The gas phase reaction:
.. math::
A \longrightarrow B + C
is carried out at 100°C in a 20 L constant-volume, sealed batch reactor, at atmospheric pressure. The reaction is second order: :math:`-r_A = k C_A^2` where :math:`k = 2\,\text{L.mol}^{-1}\text{.min}^{-1}`.
One mole of pure A is initially placed in the reactor, which is well mixed (is this a reasonable assumption?). Determine:
#. the partial pressure due to A in the reactor #. the concentration of A in the reactor after 5 minutes have elapsed #. the partial pressure due to A in the reactor after 5 minutes have elapsed.
.. answer::
There was an error in the question, so as long as you attempted the question you got full grade. The problem was over-specified, so you get a partial pressure of A that exceeds atmospheric pressure.
#. :math:`p_a V = n_A RT`, or solving for :math:`p_A = \dfrac{(1)(8.314)(373)}{20 \times 10^{-3}} = 155056\,\text{Pa}`. #. After 5 minutes in a batch reactor that is assumed to be well-mixed:
.. math:: \frac{dN_A}{dt} &= r_AV \\ \frac{dC_A}{dt} &= r_A \\ \int_{t=0}^{t}{dt} &= \int_{C_{A0}}^{C_A}{\dfrac{dC_A}{-kC_A^2}} \\ t &= \dfrac{1}{k_A}\left(\dfrac{1}{C_A} - \frac{1}{C_{A0}}\right)\\ C_A &= 0.033\,\text{mol.L}^{-1}
#. The partial pressure due to A after 5 minutes have elapsed is :math:`p_A = C_A RT = (33.0)(8.314)(373) =102337\,\text{Pa}`
.. question:: :grading: 10
Consider a municipal water treatment plant for a smallish community. Waste water at :math:`32,000\,\text{m}^3\text{.day}^{-1}`, flows through the treatment plant with a mean residence time of 8 hours. Air is bubbled through the tanks, and microbes in the tank attack and break down the organic material:
.. math::
\text{Organic waste} + \text{O}_2 \xrightarrow{\text{microbes}} \text{CO}_2 + \text{H}_2\text{O}
A typical entering feed has a BOD (biological oxygen demand) of :math:`200\,(\text{mg O}_2)\text{.L}^{-1}`, the effluent has a negligible BOD. Find the average rate of reaction, or decrease in BOD, in the treatment tanks.
.. answer::
The rate of reaction is defined as :math:`-r_{\text{O}_2} = \dfrac{\text{mol O$_2$ used}}{(\text{day})(\text{m}^3)}`.
We can obtain an estimate of the tank volume from the residence time equation: :math:`\tau = \dfrac{V}{q}` or that :math:`V = \dfrac{32000}{3} = 10667 \,\text{m}^3`.
To obtain a value for the numerator, recognize that 1 mole of waste requires 1 mole of oxygen to remove its biological oxygen demand. We require 200 mg of oxygen though per litre of waste, or :math:`\dfrac{0.2\,\text{g.L}^{-1}}{32\,\text{g.mol}^{-1}}\times 1000\,\text{L.m}^{-3} = 6.25` mol of oxygen required per :math:`\text{m}^3` of waste. There will be :math:`32 000\,\text{m}^3\text{.day}^{-1}` of waste water to treat, so this corresponds then to an average reaction rate of :math:`\dfrac{6.25 \times 32 000}{10667} = 18.75\,\dfrac{\text{mol O$_2$ used}}{(\text{day})(\text{m}^3)}`.
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