Thermo math lab 1

Specific Heat

• Can be defined as the amount of energy needed to raise a 1 kg mass by 1 Kelvin• Water has a high specific heat since it takes a relatively high amount of energy to raise 1 kg of water by 1 Kelvin

Density and Specific Volume

• Density = mass / volume• Specific volume = volume / mass • Density and specific volume are inverses of each other • Inverse is found by swapping numerator and denominator; or use 1/x function • Density times specific volume =1.

Density

• Density is mass (kg) divided by volume (m^3)• When volume is constant, density will increase as mass increases • When mass is constant, density will increase as volume decreases.

Specific Volume

• Specific Volume is volume (m^3) divided by mass (kg)• Example problem: What is the specific volume when the volume is 0.004 m^3 and the mass is 345 kg?• SV = 0.004 m^3 / 345 kg• SV = 0.000011594 m^3/kg• SV = 1.159 * 10^-5 m^3/kg

Units

Basic units: kilogram, meter, second, K• All other units are a combination of the above units• Speed: m/s = ms^-1• Acceleration: m/s^2 = ms^-2• Force: mass times acceleration = Newton = kgms^-2

Example Problem

What is the 1000 to 500 millibar thickness of air that has an average temperature of -2 degrees C and an average virtual temperature of 0 degrees C?• Solution done on overhead

Topics:

-rounding, % and decimal conversion, base and exponent calculations, powers, scientific notation, inverse function, natural log function, natural exponential function, weighted average, order of operations, equation algebra, symbols, common thermo constants, temperature conversions, units, unit reduction, derivative and integral notation

Powers

-the power is the value of the exponent• Sign of power is switched if moved from numerator to denominator or vice versa.• x^-2 = 1/x^2; x^5 = 1/x^-5• There are specific rules for multiplying and dividing powers that have like bases

Answers (1 - 5)

1. 4.89 + 7.21 + 6.762 = 18.862 = 18.9• 2. 4/7 = 0.571428571, 57%• 3. 8^1.3 = 14.9• 4. 0.000000348 = 3.48 * 10^-7• 5. inverse of 4.3 kg/m^3 = 0.23 m^3/kg

Thermo Math Lab Quiz

1. Add and round the following numbers to tenths: 4.89, 7.21, 6.762• 2. Express the fraction 4/7 as a decimal and as a percent• 3. Find value of 8^1.3• 4. Express 0.000000348 in scientific notation• 5. Give the reciprocal of 4.3 kg/m^3

Answers (11 - 14)

11. 5x/10 = 2x + 7; x/2 = 2x + 7; x = 2(2x + 7); x= 4x + 14; x - 4x = 14; -3x = 14; x = -4.667• 12. 303 K = 303 - 273 = 30 C = 86 F• 13. Pascal = N/m^2 = kgm^-1s^-2; Pascal / g = kgm^-1s^-2 / ms^-2 = kgm^-2• 14. dP/dt = 10 mb/17 hrs = 0.588 mb/hr

Thermo Math Lab quiz

6. Give value of the natural log of 50• 7. Give the value of the exponent of 1.5• 8. e^LN(2) = ?• 9. Test grade 1 is 80% and is worth 60% of grade while test grade 2 is 70% and is worth 40% of grade. What is the final grade?• 10. Solve (18/6) + 7 - (8+3) 6. LN(50) = 3.912023005• 7. e^(1.5) = 4.48168907• 8. e^LN(2) = 2• 9. 0.6(80) + 0.4(70) = 76%• 10. (18/6) + 7 - (8 + 3) = 3 + 7 - 11 = -1

Scientific Notation Examples

678,500,000,000,000.0• In scientific notation is written as 6.785 * 10^14• Decimal has been moved to the left by 14 spaces• 6.785 is called the coefficient. 10^14 is called abase 10 with an exponent of 14

Derivative Notations

A derivative is a change in a parameter. For example: Temperature is expressed as T while a change in temperature is expressed as dT. If the temperature warms from 10 C to 15 C, then the dT = +5 C• dT/dt is an expression that means a change in temperature during a change in time. If the temperature changes 5 C in 1 hour, then dT/dt = 5 C/hr• When you see the d symbol in front of a parameter it means that it is a change in that parameter

Density

Density• Example problem: The density is 3,500 kgm^-3 and the volume is 0.5 m^3. What is the mass?• D = M/V• 3,500 kgm^-3 = M/0.5 m^3• M = 3,500 kgm^-3 * 0.5 m^3• M = 1,750 kg

Air Parcels in Atmosphere

Air Parcels in Atmosphere• PV=mRT• P, V and T all change as a parcel rises or sinks adiabatically in the atmosphere. The term m remains constant • Rising parcel: Pressure decreases, Volume increases and Temperature decreases • Sinking parcel: vice versa of above

Integral Notations

An integral is used to find a quantity of a parameter between two limits• The symbol is a line with curves on both the top and bottom of the line• The integral is commonly used to find the value of a parameter that changes with a non-linear rate, thus you will often see the d and integral symbol within the same equation

Units

Area = m^2; Volume = m^3• Pressure: Force / Area = Pascal = kgms^-2 / m^2 = kgm^-1 s^-2• Density: mass / volume = kgm^-3• Specific volume: volume / mass = m^3kg^-1• Energy: Force times distance = Joule = kgm^2s^-2• Power: Energy / time = Watt = kgm^2s^-3

Equilibrium Example Problem

Assume heat is contained in a closed system. 10 kilograms of water at 1 C is poured into a 5 kilogram Copper vessel that is at 50 C. What is the temperature at thermal equilibrium of the system?• Solution done on overhead

Common Thermo Constants

Common Thermo Constants• SL pressure = 101,325 Pa = 1013.25 hPa = 1013.25 mb• Latent Heat (melting / freezing) = 3.34 * 10^5 J/kg• Latent Heat (evaporation / condensation) = 2.5 * 10^6 J/kg• Latent Heat (sublimation / deposition) = 2.83 * 10^6 J/kg• DALR = 10 C/km• Triple point of water = 0.01 C at 6.11 mb• Freezing point of water = 32 F = 0 C = 273k

Constants

Constants • Each individual gas has a unique Gas Constant (R) • Universal Gas Constant (R*): 8,314.3 J/kg/K • Gas Constant for dry air (Rd): 287 J/kg/K • Gas Constant for water vapor (Rv): 461 J/kg/K

Heat Capacity

Definition: The amount of heat required to raise the temperature of one kilogram of a substance by one degree Celsius without change of phase. Units are also commonly expressed in calories per gram.• 1 calorie = 4.186 Joules• 1 kg = 1,000 grams

Inverse Function

Density and Specific Volume are inverses of each other • 5 kg/m^3 density = 0.2 m^3/kg specific volume • On the calculator, inverse function is often denoted with a 1/x or x^-1 symbol

What is Energy?

Energy is Force that produces motion. Energy is described as the capacity to do work • Work = Force times distance• The terms Energy, Work and Heat are all describing a movement or a potential movement being produced by a Force. All have units of Joules

Avogadro's Hypothesis

Equal volumes of all gases measured at the same temperature and pressure contain the same number of molecules. This is regardless of molecular weight of the molecules

Equation Algebra

Equation Algebra• If a variable or number is added, subtracted, multiplied or divided to one side of the equation then it must be done to the other side of the equation also. For example 4x = 16. To find x, 4 must be divided by each side of the equation 4x/4 = 16/4; x = 4

Work

Example problem: A 50 Newton force pushes a couch along the floor 6 meters. How much Work is done?• W = 50 Newtons * 6 m = 300 Nm = 300 Joules

Scientific Notation

Exponent sign is positive if number is bigger than 1 and negative if number is smaller than 1 • A number in scientific notation will have one number to the left of the decimal point. This number is called the coefficient and it will have avalue between 1 and 10

temperature Conversions

F = 1.8*C + 32• C = 5/9*(F - 32)• K = C + 273• Most thermodynamic equations will require units in K since it is an absolute temperature scale• In equations where a temperature difference is needed then C or K will work

Weighted Average

For a non-weighted average, add the values and divide by the number of values• Example: What is average of 5, 9 and 11? Answer: (5 + 9 + 11) / 3 = 8.3• For a non-weighted average each term must be multiplied by the weight on each value

Equation of State and Weather

For each process explain how the variables in the equation of state will likely change for that air:• 1. Rain falling into dry air and evaporating• 2. Solar radiation warming the PBL• 3. Downsloping wind• 4. Rising air within entire troposphere

Average Molecular Weight

Found by a weighted average of the constituent gases• Example problem: Find the average molecular weight of air that is 30% Oxygen, 67% Nitrogen and 3% Water Vapor

Energy within Fluids

Gases: As energy increases the molecular motion increases. The molecules are moving faster and colliding into each other more • Liquids: Same as gases but with a tighter spacing of molecules. Molecular collisions are more of a vibration since molecules are linked to each other

Common Thermo Constants

Gravitational acceleration = 9.8 m/s^2• Universal Gas Constant (R*) = 8,314.3 J/kmol/K= 8,314.3 J/kg/K• Gas constant of dry air = Rd = 287 J K^-1 kg^-1• Gas constant of moist air = Rv = 461 J K^-1 kg^-1• Specific heat for changing Volume (V) when the Pressure (P) remains constant = Cp =1004 J/kg/K• Rd/cp = 0.286; Rd/Rv = 0.622

Heat Transfer

Heat transfer equation: H = cm (T1 - T2)H = amount of heat transferredm = massT1 = initial temperatureT2 = temperature after addition of heat c = specific heat constant

Heat

Heat• Heat is a form of energy• Definition: The transfer of energy from one body to another as a result of a difference in temperature or a change in phase

Hypsometric Equation

Hypsometric Equation• Equation is used to measure the distance between two pressure levels. This distance is known as thickness• Thickness is primarily a function of the average temperature between the two pressure surfaces• Thickness is a direct function of the average virtual temperature between the two pressure surfaces

Ideal Gas Law

Ideal Gas Law • Example problem: The air pressure is 1000 mb and the temperature is 30 C. What is the density of this dry air? • Use Rd for gas constant, temperature in gas equation must be in K, pressure must be in Pascals; 1,000 mb = 100,000 Pa

Ideal Gas Law

Ideal Gas Law• The ideal gas law relates pressure, density and temperature; P = density*Rd*T • Rd is the constant use for dry air• There are several ways the terms can be arranged in the equation and ways to combine terms to create a new term. For example, density can be broken down into volume and mass

Unit Reduction

If the same units are in the numerator and denominator then they cancel out. Example: ms^-1 / m = s^-1• If the same units are in the numerator and denominator but have a different power, then they can be reduced. Remember that the sign of the power changes when moved from numerator to denominator or vice versa. Method: move same units so they both are in the numerator. Add the powers. Example: m^-1 / m^-3 = m^-1m^3 = m^2

Log and Exponent Rules

LN(1000/500) = LN(1000) - LN(500)• e^LN(x)=x

Specific Heat of Water

Liquid Water: 1 calorie/gram; 4,190 J/kg• Ice: 0.48 calorie/gram; 2,000 J/kg• Water has a relatively high specific heat (takes a larger amount of energy to raise the temperature by 1 C)• Aluminum: 910 J/kg; Copper: 390; Lead:130 J/kg

Mass

Mass • Mass (m) is the amount of a substance • Mass is not dependent on gravity, thus mass and weight have separate definitions • Weight is mass (m) times gravity (g) • Mass can be expressed in kilograms • mass = Force / acceleration

Average Molecular Weight

Molecular weight of Oxygen (O2) = 32 atu• Molecular weight of Nitrogen (N2) = 28 atu• Molecular weight of Water Vapor (H2O) = 18 atu• 30% O2, 67% N2, 3% H2O• 0.3*(32) + 0.67*(28) + 0.03(18) = AMW• 9.6 + 18.76 + 0.54 = 28.9 at

Natural Exponential Function

Natural Exponential Function• Exponent of any number will be positive• e^(-10) = 0.0000454• e^(-5) = 0.0067379• e^(0) = 1• e^(5) = 148.413• e^(10) = 22,026.466• y = e^x; as x increases then y increases at an increasing rate

Natural Log Function

Natural Log Function • ln of 0 and a negative number is undefined• ln of a number between 0 and 1 is negative• ln(1) = 0• ln(5) = 1.6• ln(10) = 2.3 • ln(100) = 4.6• ln(1000) = 6.9• y = ln(x); As x increases the rate at which y changes decreases

order of Operation

Order of Operations• Solve operations inside the parenthesis first starting with most inner parenthesis. Next, do the multiplication and division. Last, do the addition and subtraction• Example: ((11 + 7) / (3*(6 - 3))) + 2• ((11 + 7) / (3*3)) + 2• (18 / 9) + 2 = 2 + 2 = 4

ideal Gas Law

P = density*Rd*T• T = 30 C + 273 = 303 K• 100,000 Pa = density*287 JK^-1kg^-1*303 K• 100,000 Pa = density*86,961 Jkg^-1• 100,000 Pa / 86,961 Jkg^-1 = density• Density = 1.15 kgm^-3

Density / Temperature Relationship

P=DRT• If P constant, when Density increases then Temperature decreases • There is an inversely proportional relationship between the variables D and T• c=DT • Example: Cooling of air in a hot air balloon

Density / Pressure Relationship

P=DRT• If T constant, when Density increases then Pressure increases • There is a proportional relationship between the variables D and P• P=cD • Example: pumping air into a car tire

THETA Example Problem

PT = T*(1000/P)^0.286• What is the Potential Temperature of a parcel of air at the 800 mb level that has a temperature of 10 C?• Solution done on overhead

Equation of State

PV = mRT• Terms in equation can be arranged algebraically • Pressure can be expressed as F/A • Density or specific volume can be expressed by dividing V and m terms

Equation of State in Everyday life

PV=mRT• 1. Spraying of aerosol can • 2. Refrigerator • 3. Tire Pressure• 4. Vacuum sealer

Air Parcels in Atmosphere

PV=mRT• Air parcel: It is a volume of air in the atmosphere with similar temperature and moisture characteristics throughout. The air parcel will follow thermodynamic principles as it rises or sinks• Adiabatic: Parcel is assumed to not mix with surrounding environmental air

Pressure / Temperature Relationship

PV=mRT• If m and V constant, when pressure increases then temperature increases• There is a proportional relationship between the variables P and T• P=cT• Example: Steamer, pressure cooker

Types of Energy

Potential: stored energy• Kinetic: energy of motion• Gravitational: energy by position within gravitational field• Internal: Molecular activity (temperature)

Pressure

Pressure • Pressure is the amount of Force distributed over an area • Pressure can be expressed in Pascals (Newtons per meters squared) • Air pressure depends on the mass of air above an area, assuming gravity is constant

Pressure Equation

Pressure Equation• P = density*g*h• P = 1000 kgm^-3*9.8 ms^-2 * 2 m• P = 19,600 kgm^-1s^-2 = 19,600 Pa = 196 mb

Hypsometric Equation

Pressure at base (P1) and pressure of interest (P2): Suppose you wanted to know the thickness between 700 and 850 mb, the pressure at base (P1) would be 850 mb while the pressure of top (P2) would be 700 mb. The two pressures are divided by each other then the natural log is taken of this value• LN (850/700) = 0.194156014Gravity= can be treated as a constant 9.8 ms^-2

Pressure Equation

Relates pressure, density, gravity and height• pressure = density*gravity*height• units: density = kgm^-3, gravity = ms^-2, height = m, pressure = Pascal = kgm^-1s^-2• Units of (density*g*h) = kgm^-3ms^-2m = kgm^-1s^-2 = Pascal• Gravity is a constant while the rest of the terms are variables

Rounding

Rounding simplifies numbers and it is used to express numbers in their proper uncertainty • 45.78 + 17.2896 + 23.9 = 87.0 • Round using number with the least certainty (tenths in this case) • Only round after adding all terms, 86.9696 = 87.0

Gas constant for water vapor (Rv)

Rv = R* / molecular weight of water vapor• molecular weight of water = 18.016 therefore Rv = 461 J/kg/k

Scientific Notation

Scientific Notation • Used as a simpler way to write very big and very small numbers • The number is written using a coefficient and a base 10 exponent • The exponent determines the number of spaces the decimal point is moved while the sign of the exponent determines the direction decimile point is moved

Scientific Notation Examples

Scientific Notation Examples• 0.00000000000045 • In scientific notation is written as 4.5 * 10^-13 • Decimal has been moved to the right by 13 spaces

Temperature

Temperature • Temperature is the measure of energy that relates energy to a relative hotness or coldness. It is a measure of molecular motion • Temperature can be expressed in Kelvins or Celsius • Heat is expressed in Joules (Energy). • The lowest possible temperature is 0 K (-273 C)

Equation of State

The Equation of State is derived as a combination of Charles' Law, Boyle's Law and Avogadro's hypothesis• Charles' Law: V=cT• Boyle's Law: PV=c• c is constant of proportionality that varies

The Universal Gas Constant (R*)

The Universal Gas Constant (R*) • The Universal Gas Constant (R*) is the same for any kilogram of gas. R* relates the other variables together in the equation of state. Average molecular weight is used to convert R* to R• R* = 8,314.3 J/kg/K

Gas Constant for dry air (Rd)

The apparent molecular weight of dry air is equal to 28.97 atomic units• Rd = R* / molecular weight of dry air= 287 J/kg/K

Energy Conservation

The total energy within a system must be conserved. The energy can transform to another type of energy but energy can not be created and can not be destroyed

Thermal Equilibrium Equation

Thermal Equilibrium Equation• m1*c1*(Tf - T1) + m2*c2*(Tf - T2) + mx*cx*(Tf - Tx) = 0• m = mass of substance• c = heat capacity of substance• Tf = temperature at thermal equilibrium• T = initial temperature of substances

Thermal Equilibrium

Thermal equilibrium occurs when one or more substances in a system have the same temperature• When substances have different temperatures and come in contact with each other, then heat will flow from the warmer substance toward the cooler substance• The amount of heat transferred depends on the temperature difference and heat capacities of the substance(s)

Thermo Math Concepts

Thermo Math Concepts • Purpose: Refresh math skills • Thermodynamics is a subject based on math and physics. An understanding of math concepts up to algebra is essential to understanding the mathematical concepts in thermodynamics

Thermo Math Lab 2

Thermo Math Lab 2• Topics: Ideal gas law, pressure equation, average molecular weight, density, specific volume, Heat, thermal equilibrium, Work, Poisson's equation, hypsometric equation

Inverse Function

This is also called taking the reciprocal• To find the inverse, take 1 and divide it by the original number. When dealing with units the units switch between the numerator and denominator • A number and the inverse of a number will always multiply to equal 1

Natural Exponential Function

This is used for functions that vary exponentially(rate of increase that is increasing) • Often denoted as an e^x button on the calculator • This function is used in the Clausius-Clapeyron Equation to solve for saturated vapor pressure

Transmission of Heat

Transmission of Heat • Conduction: Flow of Heat within substances that are in direct contact • Convection: Mixing of substances with different amounts of Heat • Radiation: Movement of Heat by the use of electromagnetic waves

units

Units• Lapse rate: K/km or C/km• Temperature gradient: delta T / distance = C/m• Pressure gradient: delta P / distance = (kgm^-1 s^-2 / m) = kgm^-2s^-2

Poisson's equation

Used to find potential temperature (THETA). Potential temperature is the temperature a sample of air will have when brought to the 1000 mb pressure surface at the DALR• PT = T*(1000/P)^Rd/cp = T*(1000/P)^0.286T = Temperature in KelvinsP = Pressure in millibarsRd = gas constant for dry air (287 Jkg^-1K^-1)Cp = constant pressure process (1004 Jkg^-1K^-1)

Volume / Temperature Relationships

V=mRT• if m and P constant, when Volume increases then Temperature increases • There is a proportional relationship between the variables V and T • Charles' Law: V=cT• Example: Hot air balloon

Variable Relationships

Variable Relationships• If certain terms in the equation of state are held constant, then the relationship between certain other variables can be determined

Variables

Variables• The equation of state relates many variables• These variables can be rearranged, thus there are several ways the equation of state can be written • The variables include mass (m), volume (V), density (m/V), specific volume (V/m), pressure (P), temperature (T), and constants

Volume / Pressure Relationship

Volume / Pressure Relationship• PV=mRT • If m and T constant, when Volume increases then pressure decreases • There is an inversely proportional relationship between the variables V and P • Boyle's Law: PV=c• Example: squeezing balloon

Volume

Volume• Volume is the cubic measurement of a substance • Volume can be expressed in meters cubed• Rectangular volume = length times width times height • Spherical volume = 4/3 times PI times radius cubed • Additional methods of finding volume are also available

Heat and Gases

When Heat is added to a gas it will usually expand and increase in temperature• Gases readily expand or contract when heat is added or subtracted since gas molecules are in free motion• For a given gas, an increase of Heat into the gas will generally cause it to decrease in density

Work

Work is Force times displacement• dW = Fds• dW = pdV

Base and Exponent Calculations

Written with a base and an exponent• Multiply the base by itself by the number of the exponent• 3^5 = 3*3*3*3*3 = 243• 10^3 = 10*10*10 =1000

Hypsometric Equation

Z2 - Z1 = Thickness = • ((Rd × Tv) / gravity) × LN (P1 / P2)• Rd= gas constant = 287 J K^-1 kg^-1Tv= Average virtual temperature in Kelvins between the two pressure levels (slightly higher than actual temp) Natural log = The LN button on scientific calculator

% and decimal converserson

convert a decimal to a percent multiply it by 100 and add the % sign • 0.56 = 56%; 1.23234 = 123.234%• To convert a percent to a decimal divide by 100• 33.3% = 0.333; 150% = 1.5 • Vapor pressure is 7 mb and saturation vapor pressure is 22 mb. What is RH= 100% * 7/22 = 32

Rounding

digit after digit that is being rounded to is 5 or greater then round up, if digit is 4 or less then round down• 90.327 rounds to 90.33 when rounded to hundredths • 12.6781 rounds to 12.678 when rounded to thousandth

Pressure Equation

example problem: How much pressure is applied by the water of a 2 meter deep swimming pool? • Water has a known density of 1000 kgm^-3

another example

example: quizzes are worth 10% of grade, the exercises are both 30% of grade, midterm is 30% of grade and final exam is 30% of grade. Quiz grades are: 90, 100, 90 and 90. Quiz average is 92.5%. Exercise grades are: 75, 80, 90 and 70. Exercise average is 78.75%. Midterm grade is 85% and final exam grade is 82%.• Average = 0.1*(quiz avg) + 0.3*(exercise avg) + 0.3*(midterm) + 0.3*(final)• Average = 0.1(92.5) + 0.3(78.75) + 0.3(85) + 0.3(82)• Average = 9.25 + 23.625 + 25.5 + 24.6 = 83%

Base and Exponent Calculations

if exponent is not a whole number then the use of a scientific calculator becomes imperative• 6^1.5 = 14.69693846 • Scientific calculator will have an exponent button

Results of increased Heat

temperature increase (if a sensible heat process)• Expansion (if volume can change)• Phase change (if a latent heat process)

Equation Algebra

terms that represent numbers are termed variables. For example 2x = y. x and y are variables • The sign of a variable changes when it moves from one side of an equation to the other. For example x + y = 5. This can be rewritten as y = 5 + (-x) or simply y = 5 - x

Symbols

there are many symbols used in meteorology to express constants or parameters• A list of the common symbols used in this course will be given on the overhead cameras

Multiplying and Dividing Powers

x^a * x^b = x^a+b• Example: 2^3 * 2^4 = 2^7• x^y / x^z = x^y-z• Example: 2^3 / 2^5 = 2^-2• x^-1 = 1/x; x^-2 = 1/x^2

Natural Log Function

• This is used for functions that vary logarithmically. An example is air pressure which decreases logarithmically with height • Often denoted with the LN symbol on a calculator • This function is used in the hypsometric equation