INQU 4005: Mass and Energy Balance

Homework #1: Units and Dimensional Analysis

Problem #1:

Heat capacities are usually given in terms of polynomial functions of temperature. The

equation for carbon dioxide is:

Cp = 8.4448 + 0.5757 x 10-2T – 0.2159 x 10-5T2 + 0.3059 x 10-9T3

where T is in °F and Cp is in Btu/(lb mole)( F). Convert the equation so that T can be in

°C and Cp will be in J/(g-mole)(°K).

Problem #2

A correlation for male body density, , in g/cm3 is given by the equation:

= 0.0277W-0.3 H0.725 + 0.75

where W is mass in kg and H is height in cm. What must the units of the constants

0.0277 and 0.75 be?

Problem #3

The coefficient of heat transfer between a pure saturated vapor and a cold surface can

be predicted by the following equation:

h = 0.943 1/4

where h = mean heat transfer coefficient, Btu/(hr) (ft2) ( °F)

k = thermal conductivity, Btu/(hr) (ft) ( °F)

= density, lbm/ft3

g = acceleration of gravity, 4.17 108

= enthalpy change, Btu/lbm

L = length of tube, ft

μ = viscosity, lbm/(hr) (ft)

T = temperature difference, °F

What are the units of the constant 0.943?

Problem #4

The density of a certain liquid is given by an equation of the following form:

= (A + Bt)eCP

where = density in g/cm3

t = temperature in °C

P = pressure in atm

4-a In order to maintain the dimensional homogeneity in this equation, what would the units of A, B, and C have to be?

4-b In the units above,

A = 1,096

B = 0.00086

C = 0.000953

Modify the equation so that is expressed in lbm/ft3, T in °R and P in lbf/in2.

4-c Using the equation derived in (4-b), determine the value of for T = 200°R and P = 20 lbf/in

2.

Problem #5

The velocity in a pipe in turbulent flow is expressed by the following equation:

u = k[ / ]1/2

where is the sheer stress in N/m2 at the pipe wall, is the density of the fluid in kg/m3,

u is the velocity, and k is a coefficient. Modify the equation so that the shear stress can

be introduced in lbf/ft2, and the density in lbm/ft3, so that the velocity u comes out in ft/s.

Problem #6

A correlation for male body density, , in [kg/m3] is given by the equation:

where a= 781, b = 750, W is the body mass in [kg], and H is the male height in [m].

= aW 0.3H0.725+ b

(a) Determine the units of constants a and b

(b) Modify the equation so that is in [lb/ft3]. Determine the numerical value and respective units of the new constants a’ and b’. Note that 1000 [kg/m3] = 62.43 [lb/ft3].

Table of Results

Units of a

Units of b

Modified Equation

a’ value and units

b’ value and units

Problem #7

The diffusivity coefficient of a substance can be calculated from Fick’s Law and is represented in the following equation:

D = D0e-E/1.987T

Where

D = diffusion coefficient, [cm2/s]

D0 = maximum diffusion coefficient

E = activation energy of diffusion, [cal/g-mole]

T = absolute temperature, [K]

Please answer the following:

1) What are the units of D0 and the constant 1.987? Justify your answer.

2) Derive an equation for D [in2/s] where T and E can be entered in ˚R and Btu/lb-mole, respectively.

3) What is the maximum diffusion coefficient D0 for a D = 2 in2/s, E = 1,000 Btu/lb-mole, and T = 180 ˚R.

Problem #8

Problem #9

El coeficiente de transferencia de calor (hm) para un vapor que se condensa sobre una superficie vertical se puede determinar mediante la siguiente ecuación:

donde: 1.13 constante adimensional

aceleración gravitacional densidad del líquido Calor latente de evaporación, J/kg

conductividad térmica del líquido, W/m•°C viscosidad del líquido, kg/m•s temperatura del vapor, °C

temperatura de la superficie, °C

longitud de la superficie Para que la ecuación sea dimensionalmente consistente en el sistema internacional (SI) determine:

a. (4%) Las unidades SI de g. __________________

b. (5%) Las unidades SI de . __________________

c. (10%) Las unidades SI de hm. __________________

d. (4%) Modifique la ecuación para que los términos de Tv y Tw se substituyan en kelvin (K), sin que las unidades de los demás términos cambien.

e. (5%) Si el término de se substituye en unidades de lbm/ft3, ¿cuál sería el nuevo

valor de la constante adimensonal?

f. (7%) Si el término de se substituye en unidades de °R, el término de se deja en

°C, ¿cómo se modifica la ecuación?

Problem #10

In a crystallization reactor, the governing equation is:

R (number of crystals/minute) = aD - bD2

where D = diameter of crystals, mm a and b = constants 1) Determine the units of a (5 pts) 2) Determine the units of b (5 pts)

Use values of a = 200 and b = 10 for the following questions 3) Rewrite the original equation in order to use D in inches and R in number of crystals per

second (5 pts) 4) What are the units of the two new constants resulting from part iii? (10 pts)

Problem #11

In recent years the concentration of CO 2 in the atmosphere has increased to approximately 350 ppm. Given the approximate concentration of air: O2 – 20.36% CO2 – 0.035% N2 – 76.595% Ar – 0.01% H2O – 3.0%

i) Determine the Average Molecular Weight of the mixture. ii) Determine the composition in gmole %. iii) Determine the composition in lbmole %. iv) Determine the ppm of CO2 based on mass content. v) Determine the ppm of CO2 based on mole content.

Problema #12 Una solución acuosa de HN03, contiene 10% de ácido. La temperatura es de 25 C y el peso específico (Sp. Gr.) 1.0523.

a) Si se requieren 15 kg/min de solución, el flujo volumétrico en m3/hr es: b) El peso molecular promedio de la solución es: c) La molaridad de la solución es: d) La fracción molar del ácido es:

Problem #13

Two weeks ago, the mayor of Caguas, William Miranda Marín, announced an ethanol (E85) demonstration project for his municipality. E85 is a renewable liquid fuel consisting of 85% Ethanol (CH 3CH2OH) and 15% Gasoline by volume. Assuming gasoline is mainly isooctane (C8H18), determine the following for an E85 blend: Additional information:

ethanol = 0.78 g/cm3

isooctane = 0.71 g/cm3

Assume Ideal Solution – Volumes are additive

Result Units Ethanol mass fraction kg ETOH/kg total

Ethanol mole fraction kgmoles ETOH/

kgmoles total

Ethanol Concentration kg moles Ethanol/

Liter

Isooctane Concentration kg moles Isooctane/

Liter

Mass ratio kg ETOH/ kg Isooctane

Solution Density kg total/ Liter

Problem #14

The Colburn equation for heat transfer is

where C = heat capacity, Btu/(lb of fluid)(°F); = viscosity, lb/(hr)(ft); k = thermal conductivity, Btu/(hr)(ft2)(°F)/ft; D = pipe diameter, ft; and G = mass velocity,lb/(hr)(ft2) of cross section. What are the units of the heat transfer coefficient h? Problem #15 An empirical equation for calculating the inside heat transfer coefficient, hi, for the turbulent flow of liquids in a pipe is given by:

Where hi = heat transfer coefficient, Btu/(hr)(ft)2(°F) G = mass velocity of the liquid, lbm/(hr)(ft)2 K = thermal conductivity of the liquid, Btu/(hr)(ft)(°F) Cp = heat capacity of the liquid, Btu/(lbm)(°F)

= Viscosity of the liquid, lbm/(ft)(hr) D = inside diameter of the pipe, (ft) a. Verify if the equation is dimensionally consistent. b. What will be the value of the constant, given as 0.023, if all the variables in the equation are inserted in SI units and hi is in SI units?

Problem #16 A solution of sulfuric acid at 60°F is found to have a sp gr of 1.22. From the tables in Perry's Chemical Engineer's Handbook, the solution is found to be 30 percent by weight H2SO4. What is the concentration of H2SO4 in the following units: (a) lbmol/gal, (b) lb/ft3, (c) g/L, (d) lb H2SO4/lb H2O, and (e) lb mol H2O/lb mol total solution? Problem #17

A 0.50 molar aqueous solution of sulfuric acid flows into a process unit at a rate of 1.25 m3/min. The specific gravity of the solution is 1.03. Calculate:

H2SO4 concentration kg/m3

Mass flow rate of H2SO4 kg H2SO4/s

Mass fraction of H2SO4 kg H2SO4/

kg solution

Mole ratio moles H2SO4/ moles water

Molar flow rate of H2SO4 kgmoles H2SO4/s

Problem #18 Hydrogen can be separated from natural gas by diffusion through a round tube. The rate of separation is given by:

where

N = rate of transport of H2 from the tube, gmoles/(sec)(cm of length of tube)

D = diffusion coefficient = molar density of H2, in gmoles/cm3

, where r1 and r2 are in cm.

Problem #19 A pallet of boxes weighing 10 tons is dropped from a lift truck from a height of 10 feet. The maximum velocity the pallet attains before hitting the ground is 6 ft/sec. The kinetic energy of the pallet at a maximum velocity in lbF-ft is:____________ Problem #20 A water solution contains 1.704 kg of HNO3 per kg of water, and has a specific gravity of 1.382 at 20°C. The composition of the solution expressed as:

a. weight percent HNO3 is: __________________ b. pounds HNO3 per ft3 of solution at 20°C, is: ___________________

c. molarity is: _____________________

Problem #21 The heat capacity of a substance is 5.32 J/(g)( °C) and its molecular weight is 37.4.

°

°

Problem #22 A commonly used equation for determining the volume rate of flow, Q, of a liquid though an orifice located in the side of a tank is where A is the area of the orifice, g is the acceleration of gravity, an h is the height of liquid above the orifice.

a) Is the equation dimensionally correct? explain. b) In the International System of Units, what should be the units of the numbers 0.61, and of the

number 2 under the square root symbol?. Problem #23 From Table B-2 in the Textbook, we obtain the following expression for the heat capacity of acetone as a function of temperature:

Where T is in [°C] and Cp in [kJ/(mol °C)] a) Determine the units of the constant b) Modify the equation to enter the temperature in [°F]

Problem #24 An aqueous solution containing 35 wt% of H2SO4 is available in our Lab. The solution has a specific gravity of 1.2563. Molecular weight of H2SO4 is 98.

a) Determine the molarity (moles per liter). b) Detemine the mol fraction of H2SO4 c) What volume must be taken if you want to prepare 2L of a solution 7 wt%.

Problem #25

A mixture of gases is analyzed and found to have the following composition: CO2 12.0% CO 6.0 CH4 27.3 H2 9.9 N2 44.8

How much will 3 lb moles of this gas mixture weight?

PRESSURE PROBLEMS

Problem #1 a) Dos tanques de aceite están conectados a través de un manómetro, como se muestra en la figura. La diferencia en el nivel de mercurio (Hg) en las dos ramas del manómetro es de 32”. La densidad del aceite es 45 lbm/ft3 y la densidad relativa (“gravedad especifica”) del mercurio es sp.grHg=13.55. Determine la diferencia de presión (P1-P2) en psi: _________________________ b) La presión manométrica (“gage pressure”) medida es de 65 kPa. Determine la altura de la columna de mercurio h Problem #2 A fluid of unknown densitiy is used in two manometers – one sealed end, the other across an orifice in a cooking oil pipe. See figure below. Given the following data Cooking Oil = 0.78 g/cm3 Patm = 758 mm Hg, determine the following

i) Determine the density of the manometer fluid. ii) Given P(b) = 20.0 psig, calculate P(b) in psia. iii) What is the pressure drop between point (a) and (b), in mm Hg? iv) Give your best guess of the chemical compound used in the manometer

Table of Results Question Result Units

i g/cm3 ii psia iii mmHg iv

Problem #3 Water and SAE 30 oil ( = 55.5 lb/ft3 ) flow in two pipelines. A double U-tube manometer is connecting the pipelines.

PA-PB = ________________ KPa Problem #4 Determine the elevation difference, h, between the water levels in the two open tanks shown.

Problem #5 A U-tube manometer is connected to a closed tank containing air and water as shown. At the closed end of the manometer the air pressure is 16 psia. Determine the reading on the pressure gage for a differential reading of 4 ft on the manometer. Express your answer in psi (gage). Assume standard atmospheric pressure and neglect the weight of the air columns in the manometer.

Problema 6 Un orificio se puede usar para determinar el flujo volumétrico de un gas que fluye por un conducto. Para esto, se mide la diferencia de presión a través del orificio utilizando un manómetro, tal como se ilustra en la figura: La tubería del manómetro es de 1.0 cm de diámetro interno. El volumen de la porción del

manómetro debajo del plano de referencia a–a es de 500.0 cm3.

Dato:

a. Inicialmente, el fluido del manómetro es agua, ft y la diferencia de presiones

equivale a 20.0 kPa. ¿Cuánto es en pulgadas?

b. Suponiendo que se mantiene el mismo flujo de gas, pero se extrae toda el agua del manómetro y se reemplaza por CCl4 (sp. gr. = 1.59). Con el nuevo fluido ft.

¿Cuánto sería en pulgadas?

c. Si en vez de CCl4 se utilizase mercurio (sp. gr. = 13.54), ¿cuánto sería en pulgadas?

d. Para el caso original (agua como fluido del manómetro) el flujo de gas cambia y

ft y ft. ¿Cuál es la nueva diferencia de presiones, en unidades de mm Hg?

e. Estime el volumen (cm3) de CCl4 que habría que añadir al manómetro, para el caso de la pregunta b.

Flujo de gas

a a

h2 h3

h1