Units and Dimensions

Abstract

Units used to designate magnitude of a dimension have evolved based on common usage and instruments available for measurement. Two major systems for measurement have been used: the English system, which was used primarily in industry, and the metric system, which was used in the sciences. The confusion that results from the use of various terms to represent the same dimension has led to the development of a common system of units that is proposed for use in both science and industry. The Système International d’Unites (International System of Units) and the official international designation SI was adopted in 1960 by the General Conference on Weights and Measures. This body consists of delegates from member countries of the Meter Convention, and it meets at least once every 6 years. There are at least 44 countries represented in this convention, one of which is the United States.

Problems

• 1.1. Set up dimensional equation and determine the appropriate conversion factor to use in each of the following:

  $$ {\displaystyle \begin{array}{l}\frac{\mathrm{lb}}{{\mathrm{ft}}^3}=\frac{\mathrm{lb}}{\mathrm{gal}}\times \mathrm{conversion}\ \mathrm{factor}\\ {}\frac{\mathrm{lb}}{{\mathrm{in}}^2}=\frac{\mathrm{lb}}{{\mathrm{ft}}^2}\times \mathrm{conversion}\ \mathrm{factor}\\ {}\mathrm{W}=\frac{\mathrm{cal}}{\mathrm{s}}\times \mathrm{conversion}\ \mathrm{factor}\end{array}} $$

• 1.2. The amount of heat required to change the temperature of a material from T1 to T2 is given by:

$$ \mathrm{q}={\mathrm{mC}}_{\mathrm{p}}\left({\mathrm{T}}_2-{\mathrm{T}}_1\right) $$

where q is BTU, m is mass of material in lb, C_p is specific heat of material in BTU/lb·°F, and T₁ and T² are initial and final temperatures in °F.

• (a) How many BTUs of heat are required to cook a roast weighing 10 lb from 40 °F to 130 °F? Cp = 0.8 BTU/(lb ・ °F)

• (b) Convert the number of BTUs of heat in (a) into watt-hours.

• (c) If this roast is heated in a microwave oven having an output of 200 watts, how long will it take to cook the roast?

• 1.3. How many kilowatt-hours of electricity will be required to heat 100 gallons of water (8.33 lb/gal) from 60 °F to 100 °F? C_p of water is 1 BTU/(lb ・°F).

• 1.4. Calculate the power requirements for an electric heater necessary to heat 10 gallons of water from 70 °F to 212 °F in 10 min. Express this in Joules/min and in watts. Use the following conversion factors in your calculations:

  • Specific heat of water = 1 BTU/(lb ・°F)

  • 3.414 BTU/(W ・h)

  • 60 min/h

  • 3600 s/h

  • 8.33 lb water/gal

  • 1.054 × 10³ J/BTU

• 1.5. One ton of refrigeration is defined as the rate of heat withdrawal from a system necessary to freeze 1 ton (2000 lb) of water at 32 °F in 24 h. Express this in watts and in BTU/h. Heat of fusion of water = 80 cal/g.

• 1.6. (a) In the equation τ = μ(γ), what would be the units of τ in the equation if μ is expressed in dyne ・s/cm² and γ is in s⁻¹?

• (b) If μ is to be expressed in lb_m/(ft ・s), τ is expressed in lb_f/ft², and γ is in s⁻¹, what is needed in the equation to make it dimensionally consistent?

• 1.7. In the equation

  $$ \overline{\mathrm{V}}=\frac{1000\left({\uprho}_1-{\uprho}_2\right)}{{\mathrm{m}\uprho}_1{\uprho}_2}+\frac{\mathrm{M}}{\uprho_2} $$

what units should be used for the density, ρ, such that $\overline{\mathrm{V}}$ would have the units ml/mole?

• (a) m = moles/1000 g

• (b) M = g/mole

• 1.8. Express the following in SI units. Follow the rounding-off rule on your answer.

  • (a) The pressure at the base of a column of fluid 8.325 in. high when the acceleration due to gravity is 32.2 ft/s², and the fluid density is 1.013 g/cm³:

    $$ \mathrm{P}=\mathrm{density}\times \mathrm{height}\times \mathrm{acceleration}\ \mathrm{due}\ \mathrm{to}\ \mathrm{gravity} $$

  • (b) The compressive stress (same units as pressure) on a specimen having a diameter of 0.525 in. when the applied force is 5.62 pound force:

    $$ \mathrm{Stress}=\mathrm{Force}/\mathrm{area} $$

  • (c) The force needed to restrain a piston having a diameter of 2.532 in. when a pressure of 1500 (exact) lb_f/in² is in the cylinder behind the piston:

    $$ \mathrm{Force}=\mathrm{Pressure}\times \mathrm{area} $$

• 1.9. An empirical equation for heat transfer coefficient in a heat exchanger is:

  $$ \mathrm{h}=\mathrm{a}{\left(\mathrm{V}\right)}^{0.8}\left(1+0.011\mathrm{T}\right) $$

where h = BTU/(h ・ft² ・Δ°F), V = ft/s, and T = °F. In one experimental system, a had a value of 150. What would be the form of the equation and the value of a if h, V, and T are in SI units?

• 1.10. A correlation equation for the density of a liquid as a function of temperature and pressure is as follows:

  $$ \mathrm{d}=\left(1.096+0.0086\ \mathrm{T}\right){{{\left(\mathrm{e}\right)}^{0.}}^{000953}}^{\mathrm{P}} $$

where d is density in g/cm³, T is temperature in Kelvin, P is pressure in atm, and e is the base of natural logarithms. A normal atmosphere is 101.3 kPa. Determine the form of the equation if all variables are to be expressed in SI.

• 1.11. The Arrhenius equation for the temperature dependence of diffusivity (D) is given by D = D_o[e]^−E/RT. R is a constant with a value of 1.987 cal/mole K. (T is temperature in degrees Kelvin). If D is in cm²/s, determine the units of D_o and E.

• 1.12. The heat of respiration of fresh produce as a function of temperature is q = a e^bT. If q has units of BTU/(ton・24 h), and T is in °F, determine the units of a and b. The values of a and b for cabbage are 377 and 0.041, respectively. Calculate the corresponding values if q is expressed in mW/kg and T is in °C.