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Units and Dimensions

SEPTEMBER 30, 1999

Likely Cause Of Orbiter Loss Found

The peer review preliminary findings indicate that one team used English units (e.g., inches, feet and pounds) while the other used metric units for a key spacecraft operation.

Mars Climate Orbiter

Quantity = numerical value & units

Example - 100 kg/hr

Dimensions = basic concepts of measurement

Units = quantitatively expressing dimensions

All dimensions of interest can be expressed in terms of:

Mass

Length

Temperature

Time

Currency

What are the dimensions & (SI units) on the following ?

velocity =

acceleration =

force =

pressure =

energy =

L/t (m/sec)

L/t2 (m/sec2 )

M*L/t2 (Kg m/ sec2)

M/L*t2 (Kg m / m2 sec2)

M*L2/t2 Kg m2 / sec2)

Typically, coefficients in physical laws (eg, KE = ½ mv2),exponents, and arguments (log x, sin x, exp x = ex) have no dimensions.

There are special dimensionless numbers used in chemical

engineering; for example:

Reynolds Number

Prandtl Number

Dimensional Homogeneity & Dimensionless Numbers

• every added and subtracted term in any equation must have the same dimensions.

Multiplication & Division of quantities

• creates compound dimensions and units

Addition and Subtraction of quantities

• must have same dimensions & units

Example

Consider the equation D(ft) = 3t(s) + 4

• What are the dimensions and units of 3 and 4 ?

Convert the equation D(ft) = 3t(s) + 4 to D’(m) = __t’(min) + __

Convert each term then substitute ...

D(ft) = D’(m) * 3.2808 ft / m & t(s) = t’(min) 60 s / min

Thus, 3.2808D’(m) = 3*[60 t’(min)] + 4

D’(m) = 55t’(min) + 1.22

• What are the dimensions & units of 55 and 1.22 ?

Example

You are traveling at 51 km/hr and increase your speed

by 1 ft/s; what is your new velocity?

Can you add these because they have the same dimensions ?

Dimensional ledger/ equations

• think units first, then numerical values

• break big problem down

10 Minute Problem

- 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)
- μ = Viscosity of the liquid, lbm/(ft)(hr)
- D = inside diameter of the pipe, (ft)
- 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.

Extra Practice Problems

Problem Set Handout: I-1 – I-17

Mass, Weight, and Force

Mass: amount of material - mass ≠ weight

Weight: Force that material exerts due to gravity (g) which changes with location, etc.

Force: (Newton, dyne, or lbf) = mass * acceleration (F = m *a)

Mass = kg (SI), g (CGS), or lbm (English)

10 Minute Problem

Momentum (lbf) is equal to mass (lbm / sec) X velocity (ft/sec)

Determine the momentum force transferred to a wall by a stream of

water flowing from a fire hose at 50 ft/sec and 1000 lb/hr.

Extra Practice Problems

Problem Set Handout: I-18 – I-21

Moles

Mole = certain number of entities

6.023 X 1023 molecules

• g-mole = amt of substance whose mass in grams is

equal to the molecular weight of the substance

• similarly kg-mole & lb-mole

• molecular weight (MW) =

• atomic weight - atomic mass .... Inside back cover of textbook

10 Minute Problem

Silver nitrate (lunar caustic) is a white crystalline salt, used in marking inks, medicine and chemical analysis. How many kilograms of silver nitrate (AgNO3) are there in :

a. 13.0 lb mol AgNO3.

b. 55.0 g mol AgNO3

Calcium carbonate is a naturally occurring white solid used in the manufacture of lime and cement. Calculate the number of lb mols of calcium carbonate in:

a. 50 g mol of CaCO3.

b. 150 kg of CaCO3.

c. 100 lb of CaCO3.

Density, Specific Gravity, API Gravity

- Density = r [=] M/L3 → kg/m3, lbm / ft3, g/cc, etc.
- • r ≠ constant → f(T,P)
- Specific volume = V = volume / unit mass = r–1 [=] L3/M
- Specific gravity = sp gr = SG =
- • For liquids & solids: rref = H2O(liquid) at 4°C & 1 atm
- [rwater= 1 g/cm3 = 1000 kg/m3 = 62.43 lbm/ft3]
- For gases: rref = air at “standard conditions”

Example: SG of Ethanol at 140 F

Example

The density of a liquid is 1500 kg/m3 at 20°C.

- What is the specific gravity 20°C/4°C of this

material ?

- What is the API gravity of the liquid ?
- What volume (ft3) does 140 lbm of this material

occupy at 20°C ?

Composition

- Mole fraction =
- Mass fraction =
- Volume fraction (gas) ????

Example

A liquefied mixture of n-butane, n-pentane and n-hexane has the following composition in weight percent.

n - C4H10 = 50 %

n - C5H12 = 30 %

n - C6H14 = 20 %

Calculate the weight fraction, mol fraction and mol percent of each component and also the average molecular weight of the mixture.

10 Minute Problem

A mixture of gases is analyzed and found to have the

following composition (volume percent). How much will

3 lb mol of this gas weigh ?

CO2 12.0

CO 6.0

CH4 27.3

H2 9.9

N2 44.8

Total 100.0

Concentration

Concentration = quantity of A / volume

kg / m3 kg mol / m3 g/L g /cc

lb / ft3 lb mol / ft3

Example

A solution of HNO3 in water has a specific gravity of 1.10 at 25 C. The concentration of HNO3 is 15 g/L.

What is the mole fraction of HNO3 in the solution ?

What is the ppm (wt) of HNO3 in the solution ?

10 Minute Problem

The 1993 Environmental Protection Agency (EPA) regulation contains standards for 84 chemicals and minerals in drinking water. According to the EPA one of the most prevalent of the listed contaminants is naturally occurring antimony. The maximum contaminant level for antimony and nickel has been set at 0.006 mg/L and 0.1 mg/L respectively.

A laboratory analysis of your household drinking water shows the antimony

concentration to be 4 ppb (wt) (parts per billion) and that of nickel to be 60 ppb (wt).

Determine if the drinking water is safe with respect to the antimony and nickel levels. Assume density of water to be 1.00 g/cm3

Extra Practice Problems

Problem Set Handout: I-22 – I-44

Temperature

Temperature - average kinetic energy of molecules.

Relative

Fahrenheit (°F)

Celsius (°C)

Absolute

Rankin ( °R )

Kelvin (°K)

Example

DT ≠ T - conversions approaches are different

Given the following equation:

- Where: r [=] gm / cm3, T [=] °C, P [=] atm
- Determine the units on the three constants
- Convert the constants to accurately reflect the following
- revised set of units:
- r [=] lbm / ft3, T [=] °R, P [=] psi

Extra Practice Problems

Problem Set Handout: I-45 – I-51

Pressure

Pressure is defined as the amount of force exerted on a unit area of a substance:

Direction of fluid pressure on boundaries

Furnace duct

Pipe or tube

Heat exchanger

Pressure is a Normal Force

(acts perpendicular to surfaces)

It is also called a Surface Force

Dam

Standard Atmosphere

1 Atmosphere

33.91 ft of water (ft H20)

14.696 psi (lbf / in2)

29.92 in Hg

760 mm Hg

1.013 X 105 Pascal (Pa)

101.3 kPa

Pressure distribution for a fluid at rest

- Let’s determine the pressure distribution in a fluid at rest in which the only body force acting is due to gravity
- The sum of the forces acting on the fluid must equal zero

Pressure distribution for a fluid at rest

A force balance in the z direction gives:

For an infinitesimal element (Dz0)

Incompressible fluid

Liquids are incompressible i.e. their density is assumed to be constant:

When we have a liquid with a free surface the pressure P at any depth below the free surface is:

Po is the pressure at the free surface (Po=Patm)

By using gauge pressures we can simply write:

Measurement of Pressure Differences

Apply the basic equation of static fluids to both legs of manometer, realizing that P2=P3.

Example

A U-tube manometer is used to determine the pressure drop across an orifice meter. The liquid flowing in the pipe line is a sulfuric acid solution having a specific gravity (60°/60°) of 1.250. The manometer liquid is mercury, with a specific gravity (60°/60°) of 13.56. The manometer reading is 5.35 inches, and all parts of the system are at a temperature of 60°F.

What is the pressure drop across the orifice meter in psi ?

10 Minute Problem

The barometric pressure is 720 mm Hg. The density of the oil is 0.80 g/cm3 . The density of mercuryis 13.56 g/cm3 The pressure gauge (PG) reads 33.1 psig. What is the pressure in kPa of the gas ?

3 in

Gas

12 in

20 in

24 in

16 in

3 in

PG

Extra Practice Problems

Problem Set Handout: I-52 – I-63

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