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2.7: Problems

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  • Problems marked with the symbol \(\ddagger\) will be difficult to solve without the use of computer software.

    Section 2.1

    1. The following prefixes are officially approved for various multiples of ten:

    \[\begin{aligned} 10 \to \text{deca} \to \text{D} , && 100 \to \text{ hecto} \to \text{ H} , && 1000 \to \text{ kilo} \to \text{ K} \\ 10^6 \to \text{ mega} \to \text{ M} , && 10^{9} \to \text{ giga} \to \text{ G} , && 10^{12} \to \text{ tera} \to \text{ T} \end{aligned}\]

    How would you express the following quantities in terms of a prefix and a symbol using the appropriate SI unit?

    (a) 100,000,000 watt (b) 100 meter (c) 300,000 meter
    (d) 100,000 hertz (e) 200,00 kg (f) 2,000 ampere

    2. If a 1-inch nail has a mass of 2.2 g, what will be the mass of one mole of 1-inch nails?

    Section 2.2

    3. Convert the following quantities as indicated:

    a) 5000 cal to Btu

    b) 5000 cal to watt-s

    c) 5000 cal to N-m

    4. At a certain temperature, the viscosity of a lubricating oil is \(0.136 \times 10^{-3}\) \(lb_f\) \(s/ft^2\). What is the kinematic viscosity in \(m^2/s\) if the density of the oil is \(\rho = 0.936\) \(g/cm^{3}\).

    5. The density of a gas mixture is \(\rho = 1.3\) \(kg/m^{3}\). Calculate the density of the gas mixture in the following list of units:

    (a) \(lb_m/ft^{3}\)

    (b) g/\(cm^3\)

    (c) g/L

    (d) \(lb_m/in^{3}\)

    6. Write an expression for the volume per unit mass, \(\hat{V}\), as a function of the molar volume, \(\tilde{V}\), (that is the volume per mole) and the molecular mass, \(MW\). Write an expression for the molar volume, \(\tilde{V}\), as a function of the density of the component, \(\rho\), and its molecular mass, \(MW\).

    7. The CGS system of units was once commonly used in science. What is the unit of force in the CGS system? Find the conversion factor between this unit and a Newton. Find the conversion factor between this unit and a \({lb}_{f}\).

    8. In rotating systems one uses angular velocity in radians per second. How do you convert revolutions per minute, rev/min, to rad/s?

    9. Platinum is used as a catalyst in many chemical processes and in automobile catalytic converters. If a troy ounce of platinum costs $100 and a catalytic converter has 5 grams of platinum, what is the value in dollars of the platinum in a catalytic converter? To solve this problem one needs to determine the relation between a troy ounce and a gram, and this conversion factor is not listed in Table \(2.2.3\). While the troy ounce originated in sixteenth century Britain, it has largely been replaced by the measures of mass indicated in Table \(2.2.3\). However, it is still retained today for the measure of precious stones and metals such as platinum, gold, etc.

    10. In the textile industry, filament and yarn sizes are reported in denier which is defined as the mass in grams of a length of 9000 meters. If a synthetic fiber has an average specific gravity of 1.32 and a filament of this material has a denier of 5.0, what is the mass per unit length in pounds-mass per yard? What is the cross sectional area of this fiber in square inches? The specific gravity is defined as

    \[ \text{specific gravity} = \frac{ \text{density}}{ \text{density of water}}\nonumber\]

    11. (Adopted from Safety Health and Loss Prevention in Chemical Processes by AIChE). The level of exposure to hazardous materials for personnel of chemical plants is a very important safety concern. The Occupational Safety and Health Act (OSHA) defines a hazardous material as any substance or mixture of substances capable of producing adverse effects on the health and safety of a human being. OSHA also requires the Permissible Exposure Limit, or PEL, to be listed on the Material Safety Data Sheet (MSDS) for the particular component. The PEL is defined by the OSHA authority and is usually expressed in volume parts per million or ppm. Vinyl chloride is believed to be a human carcinogen, that is an agent which causes or promotes the initiation of cancer. The PEL for vinyl chloride in air is 1 ppm, i.e., one liter of vinyl chloride per one million liters of mixture. For a dilute mixture of a gas in air at ambient pressure and temperature, one can show that that volume fractions are equivalent to molar fractions. Compute the PEL of VC in the following units:

    (a) moles of VC/\(m^3\)

    (b) grams of VC/\(m^3\)

    (c) moles of VC/mole of air

    12. This problem is adopted from Safety Health and Loss Prevention in Chemical Processes by the AIChE. Trichloroethylene (TCE) has a molecular mass of 131.5 g/mol so the vapors are much more dense than air. The density of air at 25 C and 1 atm is \(\rho_{air} = 1.178\) \(kg/m^{ 3}\), while the density of TCE is \(\rho_{ TCE} = 5.37\) \(kg/m^{ 3}\). Being much denser than air, one would expect TCE to descend to the floor where it would be relatively harmless. However, gases easily mix under most circumstances, and at toxic concentrations the difference in density of a toxic mixture with respect to air is negligible. Assume that the gas mixture is ideal (see Sec. \(5.1\)) and compute the density of a mixture of TCE and air at the following conditions:

    (a) The time-weighted average of PEL for 8 hours exposure, \(\rho_{mix} = 100\) ppm.

    (b) The 15 minute ceiling exposure of \(\rho_{mix} = 200\) ppm.

    (c) The 5 minute peak exposure of \(\rho_{mix} = 300\) ppm.

    13. A liquid has a specific gravity of 0.865. What is the density of the liquid at 20 C in the following units:

    (a) kg/\(m^3\)

    (b) lbm/\(ft^{3}\)

    (c) g/\(cm^3\)

    (d) kg/L

    Section 2.3

    14. In order to develop a dimensionally correct form of Equation \((2.3.1)\), the appropriate units must be included with the numerical coefficients, 1920 and 597. The units associated with the first coefficient are given by Equation \((2.3.3)\) and in this problem you are asked to find the units associated with 597.

    15. In the literature you have found an empirical equation for the pressure drop in a column packed with a particular type of particle. The pressure drop is given by the dimensionally incorrect equation

    \[\Delta p = 4.7 \left(\frac{\mu^{0.15} H \rho^{0.85} { v}^{ 1.85} }{d_{p}^{1.2} } \right)\nonumber\]

    which requires the following units:

    \(\Delta p\) = pressure drop, \({ lb}_{ f} /{ ft}^{2}\)

    \(\mu\) = fluid viscosity, \(lb_m\)/ft s

    \(H\) = height of the column, ft

    \(\rho\) = density, \(lb_m/ft^3\)

    \(v\) = superficial velocity, ft/s

    \(d_{p}\) = effective particle diameter, ft

    Imagine that you are given data for \(\mu\), \(H\), \(\rho\), \(v\) and \(d_{p}\) in SI units and you wish to use it directly to calculate the pressure drop in \({ lb}_{ f} /{ ft}^{2}\). How would you change the empirical equation for \(\Delta p\) to obtain another empirical equation suitable for use with SI units? Note that your objective here is to replace the coefficient 4.7 with a new coefficient. Begin by putting the equation in dimensionally correct form, i.e., find the units associated with the coefficient 4.7, and then set up the empiricism so that it can be used with SI units.

    16. The ideal gas heat capacity can be expressed as a power series in terms of temperature according to

    \[C_{p} = A_{ 1} + A_{ 2} T + A_{ 3} T^{2} + A_{ 4} T^{3} + A_{ 5} T^{4}\nonumber\]

    The units of \(C_{p}\) are joule/(mol K), and the units of temperature are degrees Kelvin. For chlorine, the values of the coefficients are: \(A_{1} = 22.85\), \(A_{ 2} = 0.06543\), \(A_{ 3} = -1.2517\times 10^{-4}\), \(A_{ 4} = 1.1484\times 10^{-7}\), and \(A_{ 5} = -4.0946\times 10^{-11}\). What are the units of the coefficients? Find the values of the coefficients to compute the heat capacity of chlorine in cal/g C, using temperature in degrees Rankine.

    Section 2.4

    17. A standard cubic foot, scf, of gas represents one cubic foot of gas at one atmosphere and 298.15 K. This means that a standard cubic foot is a convenience unit for moles. This is easy to see in terms of an ideal gas for which the equation of state is given by (see Sec. \(5.1\))

    \[pV = n R T \nonumber\]

    The number of moles in one standard cubic foot of an ideal gas can be calculated as

    \[n = pV /RT \begin{cases} p = \text{ one atmosphere} \\ V = \text{ one cubic foot} \\ T = 298.15 \ K \end{cases} \nonumber\]

    and for a non-ideal gas one must use an appropriate equation of state14. In this problem you are asked to determine the number of moles that are equivalent to one scf of an ideal gas (see Sec. \(5.1\)).

    18. Energy is sometimes expressed as \({ v}^{ 2} /2g\) although this term does not have the units of energy. What are the units of this term and why would it be used to represent energy? Think about the fact that \(\rho gh\) represents the gravitational potential energy per unit volume of a fluid and that \(h\) is often used as a convenience unit for gravitational potential energy. Remember that \(\frac{1}{2} \rho { v}^{ 2}\) represents the kinetic energy per unit volume where \(v\) is determined by \({ v}^{2} = \mathbf{v}\cdot \mathbf{v}\) and consider the case for which the fluid density is a constant.

    Section 2.5

    19. Find the dimensions of the following product

    \[Re = \frac{\rho D { v}}{\mu }\nonumber\]

    in which \(\rho\) is the density of a fluid, \(D\) is the diameter of a pipe, \(v\) is the velocity of the fluid inside the pipe, and \(\mu\) is the viscosity of the fluid.

    20. A useful dimensionless number used in characterization of gas-liquid flows is the Weber number, defined as

    \[\text{We} = \frac{D_{b} U_{b}^{2} \rho }{\sigma }\nonumber\]

    where \(\rho\) is the density of the fluid, \(D_{b}\) is the diameter of a bubble, \(U_{b}\) is the velocity of the bubble with respect to the surrounding liquid, and \(\sigma\) is the interfacial gas-liquid tension. Verify that the Weber number is dimensionless.

    21. Given a gas mixture consisting of 5 \({ lb}_{ m}\) of methane, 10 \({ lb}_{ m}\)of ethane, 5 \({ lb}_{ m}\)of propane, and 3 \({ lb}_{ m}\) of butane, determine the number of moles of each component in the mixture.

    Section 2.6

    22. Given the following \(3\times 3\)matrices

    \[\mathbf{ A} = \begin{bmatrix} {-3} & {5} & {-4} \\ {6} & {1} & {9} \\ {4} & {-3} & {2} \end{bmatrix} \quad \mathbf{ B} = \begin{bmatrix} {2} & {-1} & {3} \\ {1} & {2} & {5} \\ {-3} & {-5} & {2} \end{bmatrix}\nonumber\]

    determine \(\mathbf{ A}+\mathbf{ B}\) and \(\mathbf{ A}-3 \mathbf{ B}\).

    23. Given the following \(1\times 4\) row matrices

    \[\mathbf{ a} = \begin{bmatrix} {3} & {1} & {5} & {6} \end{bmatrix}, \quad \mathbf{ b} = \begin{bmatrix} {6} & {-2} & {0} & {4} \end{bmatrix}\nonumber\]

    determine \(2\mathbf{ a}+\mathbf{ b}\).

    24.\(\ddagger\) Compute the \(4\times 4\) matrix defined by \(\mathbf{ A}-\mathbf{ B}+\mathbf{ C}\) where \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are given by

    \[\mathbf{ A} = \begin{bmatrix} {0.856} & {0.328} & {0.0663} & {0.908} \\ {0.529} & {0.426} & {0.364} & {0.434} \\ {0.506} & {0.812} & {0.400} & {0.137} \\ {0.652} & {0.402} & {0.0276} & {0.995} \end{bmatrix} \quad \mathbf{ B} = \begin{bmatrix} {1.142} & {0.438} & {0.0884} & {1.211} \\ {0.705} & {0.568} & {0.485} & {0.579} \\ {0.675} & {1.083} & {0.534} & {0.183} \\ {0.869} & {0.535} & {0.368} & {1.326} \end{bmatrix}\nonumber\]

    \[\mathbf{ C} = \begin{bmatrix} {0.285} & {0.109} & {0.0221} & {0.303} \\ {0.176} & {0.142} & {0.121} & {0.145} \\ {0.169} & {0.271} & {0.133} & {0.0457} \\ {0.217} & {0.134} & {0.0092} & {0.332} \end{bmatrix}\nonumber\]

    25.\(\ddagger\) Consider a project in which all the observables are given in CGS units. The project specifications require that all calculations be done in S.I. units. Define a conversion array as a set of replacement rules to assist in the conversion of units. Test your conversion array on the following quantities:

    Thermal conductivity: \(\mathbf{k} = \mathbf{65.1}\) \(\mathbf{g}\) \(\mathbf{cm}\) \(\mathbf{s^{-3}}\) \(\mathbf{K^{-1}}\)

    Heat transfer coefficient: \(\mathbf{h} = \mathbf{124.8}\) \(\mathbf{g}\) \(\mathbf{cm^{2}}\) \(\mathbf{s} \)

    26.\(\ddagger\) Write a set of replacement rules that will allow you to express the thermal conductivity in Problem 25 in terms of W \(m^{ -1}\) \(K^{ -1}\).

    1. In the Sacramento Bee, November 11, 1999 one finds the headline, Training faulted in loss of $125 million Mars probe, and in the article that follows one reads, “ The immediate cause of the spacecraft’s Sept. 23 disappearance as it entered Mars orbit was a failure by a young engineer… make a simple conversion from English units to metric…..” ↩

    2. Hurley, J.P. and Garrod, C. 1978, Principles of Physics, Houghton Mifflin Co., Boston. ↩


    4. Feynman, R.P., Leighton, R.B. and Sands, M. 1963, The Feynman Lectures on Physics, Vol. I, Addison-Wesley Pub. Co., New York.↩

    5. Sometimes chemical engineers make use of the “pound-mole” as a unit of measure; however, in this text we will be consistent with chemists, physicists and biologists and use only the mole as a unit of measure.↩

    6. For a more extensive list of prefixes see↩

    7. Rouse, H. and Ince, S. 1957, History of Hydraulics, Dover Publications, Inc., New York.↩

    8. See Truesdell, C. 1968, Essays in the History of Mechanics, placeCitySpringer-Verlag, StateNew York.↩

    9. ↩

    10. Definition: Depending on experience or observation alone, without due regard to science and theory.↩

    11. Perry, R.H., Green, D.W. and Maloney, J.O., 1984 Perry’s Chemical Engineers’ Handbook, 6\(^{th}\) Edition, McGraw-Hill, Inc. New York.↩

    12. Arrays will be represented using the font GoudyHandtooledBT with the hope that this font can be accurately reproduced for publication. In 10 point font, the printed version shown here is not very distinctive.↩

    13. Here, and throughout the entire text, we used Arial font to represent matrices.↩

    14. See for example, Sandler, S.I. 2006, Chemical, Biochemical, and Engineering Thermodynamics, 4th edition, John Wiley and Sons, New York↩

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