Heat

1. Heat • First Law of Thermodynamics: • The change in internal energy of a closed system, DU, is given by: where, DQ = heat added to the system, DW = work done by the system. • Heat is one form of energy. It is energy transferred between systems due to a change in temperature.

2. Lagrangian systems: An air parcel is studied as it moves and changes in the atmosphere. • Eulerian systems: Air at one location is studied to see how it changes at that one location. One must consider the transport in and out of the air parcel.

3. Sensible Heat: Heat that can be sensed by humans. It is that portion of total heat associated with a temperature change. Sometimes it called enthalpy (heat function). Sensible heat per unit mass is given by:

4. Cp is the specific heat at constant pressure of the material, the amount of energy necessary to change the temperature of one unit mass of material by 1o (usually oC or oK). • For dry air: • For moist air: where, r = mixing ratio of the air. (gvapor/gdry air) (Sometimes given as gvapor/kgdryair) • For liquid water, as rain or cloud droplets, the specific heat is: It varies with the temperature.

5. Latent Heat: Heat absorbed or released per unit mass by a system during a change of phase. Temperature does not measure the entire internal energy of a substance, only the translational kinetic energy part. Phase changes occur with no change in temperature. • Amount of heat energy per unit mass released or absorbed by water molecules during a phase change is defined as: • Values of latent heat in text, pg. 44, are at 0oC.

6. Gas Water Ice

7. ProcessGains EnergyLoses Energy Evaporation Water mol. Environment (vaporizaton) Melting Water mol. Environment Sublimation Water mol. Environment Condensation Environment Water molecule Freezing Environment Water molecule (fusion) Deposition Environment Water molecule

8. Specific Heat: (heat capacity) Amount of heat energy required to change the temperature of one unit mass (usually kg or gram) of a substance by 1o (usually oC or oK). For water the specific heat is not constant. Specific heat for water varies from 4177.5 to 4216.89 J/kg oC. Also, it varies for ice and steam.

9. Lagrangian Heat Budget - Part 1 • Consider only “dry” air parcels - there is no phase changes of water occurring. • First Law of Thermodynamics: The change in temperature results from heat energy being added or removed from a volume of air, and/or the volume does work - expands or contracts when rising or sinking. Then, where, Cp = specific heat, r = density • Cp Dt = sensible heat change, DP/r = work done

10. Using the hydrostatic equation: then, Substituting gives: Rearranging gives: and, eq. 3-5

11. Lapse Rate • Lapse Rate: The rate of change of temperature with height. • Types: • Environmental - Actual change as measured. • Adiabatic - How a parcel of air changes as it moves vertically in the atmosphere. No heat energy is added or removed from the parcel. • Dry adiabatic change - no phase change is occurring. • Moist adiabatic change - phase change is occurring

12. In the dry adiabatic process: and,

13. which gives: or: Dry adiabatic lapse rate, Note: temperature decreases as parcel ascends and increases as parcel descends.

14. In terms of pressure. Written in derivative form: Substituting from General Gas Law Equation. Integrating from P1 and T1 to P2 and T2

15. Integrating from P1 and T1 to P2 and T2

16. Since, • Then taking the antilog of both sides gives: For dry air, Rd/Cp = 0.28571 • This shows that moving a parcel from a pressure P1 to P2 will change the temperature from T1 to a new value given by (assuming no phase changes occur):

17. Potential Temperature • Potential temperature: The temperature a parcel of air would have if brought adiabatically (no heat transfer into or out of) from its initial state to an arbitrarily selected height or pressure level. The standard pressure normally used is 1000 hPa (100 kPa). • For height coordinates: remember:

18. Lapse rate in height units is: This can be written as: If we let T2 be defined as the Potential Temperature, Q, (the temperature the parcel would have if we moved it vertically to a particular height), then: and, Note: The solved problem, pg. 47, the parcel is descending from a height of 500 m to a height of 0 meters, ground level or sea level. Dz = 500 m

19. Lapse rate in pressure units: If we let T2 be the potential temperature, Q, then: Typically, in pressure units, the air parcel is moved to 1000 hPa (100 kPa). • Remember: This is for moving “dry” air parcels, (unsaturated, no liquid water or ice droplets), in which no energy is added or removed to the parcel.

20. Remember: Virtual temperature is given by the equation: • If there is water vapor, liquid water droplets, the potential temperature should take their effects into consideration. The following virtual potential temperature equations should be used. For non-cloudy skies: (water vapor only) • Where r = mixing ratio. Note: must be in units of g/g.

21. For cloudy skies: (liquid droplets and/or ice crystals) where, rs = saturation mixing ratio, rL = mixing ratio of liquid water.

22. Thermodynamic Diagrams • Pressure • Temperature • Dry Adiabats

23. Eulerian Heat Budget • Consider a fixed volume. The change in heat of the volume will change the temperature. That heat change may be due to several processes: (1) movement of air of different temperature into or out of the volume, (2) radiation, (3) conduction from surfaces; i.e., ground, buildings, etc., (4) turbulence (a mixing of air). Also, internal changes such as evaporation, condensation, (from latent heat); chemical reactions.

24. The change in temperature with time can be written in kinematic form as changes due to fluxes moving air and due to internal changes as: where, represents flux gradients of heat into or out of the volume (i.e., there is convergence into or divergence from the volume).(Dx, Dy, Dz are lengths of sides of the volume • represents changes in internal sources of heat.

25. Each of the change in flux terms is due in part to advection, conduction, turbulence and radiation. • Advection: The process of transport of an atmospheric property by the motion of the atmosphere (wind). • Conduction: Transfer of energy within and through a conductor by means of internal particle or molecular activity.

26. Turbulence: A state of fluid flow in which the instantaneous velocities exhibit irregular and apparently random fluctuations. • Radiation: The process by which electromagnetic radiation is propagated through free space by virtue of joint undulatory variations in the electric and magnetic field in space.

27. Advection • In the horizontal, the flux gradient due to advection is: • In the vertical, the change in temperature due to adiabatic changes must be included.

28. Conduction and Surface Fluxes • Conduction within a volume of air. Most heat transfer by conduction is negligible. • Conduction at the surface of the earth is of primary importance. • Turbulence is important in the Atmospheric Boundary Layer (lower 1 - 2 km), and in unstable environments. • Not important near the surface - becomes wind speed is zero at ground level.

29. Effective Surface Turbulent Heat Flux • Within Atmospheric Boundary Layer effects are often combined in an Effective Surface Turbulent Heat Flux. • Given by empirical formula for strong horizontal advection (windy days): where, M = mean wind speed at 10 meters elevation. CH = bulk heat transfer coefficient (smooth surface: 2 x 10-3, rough 2 x 10-2) Tair = temperature at 10 meters Tsfc = temperature at ground level

30. During calm days with strong radiational heating, there is more thermals occurring, greater vertical mixing. Then, a better equation for the Effective Surface Turbulent Heat Flux is: or: where, qML = potential temp. at height of 500 m bH = convective transport coefficient = 5 x 10-4 aH = mixed-layer transport = 0.0063 wB=Buoyancy velocity scale • w*= Deardorff velocity

31. Wind Chill • Wind Chill: The air temperature in calm conditions that would produce the same flux (rate of heat loss) by unclothed skin in shady conditions that the environmental air temperature and wind speed produce. • Note: NWS and Canada have adopted a new Wind Chill Formula.

32. Wind Chill • The environmental wind speed is assumed to be at least 1.34 m/s (average walking speed = 3 mph). • Actual heat flux varies with different persons under the same environmental conditions. • One version of the wind chill equation is: • As shown, there must be a difference in temperature - skin to air- in order for energy to leave the skin.

33. Even in calm winds there should be a wind chill value due to the movement of the person. • New NWS formula: WC in oF • T in oF V, wind speed, in miles/hour • Table 3-1 and Figure 3:10 are no longer used by NWS.

34. Heat Index • An attempt to quantify the “sultriness” of a set of environmental conditions and represents the extent to which humidity aggravates the physiological effects of high temperature on the human body. • Since, during high temperature conditions the body tries to get rid of heat, in part, by sweating, that ability is reduced, the internal heat of the body increases and the person is in danger of physiological damage; such as heat stroke, heat cramps, heat exhaustion, death. • Stull uses the 16-Element Heat Equation.

35. There are several equations used to calculate the Heat Index. • Stull’s 16-element equation: • NWS Southern Region Technical Attachment, SR/SSD 90-23, Fort Worth, TXfrom the equation by Lans P. RothfuszT in oF and Relative Humidity, R, in %HI = -42.379 + 2.04901523T + 10.14333127R - 0.22475541TR - 6.83783x10 -3 T 2 - 5.481717x10 -2 R 2 + 1.22874x10 -3 T 2R + 8.5282x10 -4 TR 2 - 1.99x10 -6 T 2 R 2 • USAToday: T in oF, Relative Humidity, RH, in % Heat index (HI)= - 42.379 + 2.04901523(T) + 10.14333127(RH) - 0.22475541(T)(RH) - (6.83783x10-3)(T2) - (5.481717x10-2)(RH2) + 1.22874x10-3)(T2)(RH) + 8.5282x10-4)(T)(RH2) - 1.99x10-6)(T2)(RH2)

36. Problems • N1, N2, N3, N5(a, d, g), N6 (a, d, g), N7 (a, e, g), N8 (a, b, c), N9 (a, b), N16 (b, d, f) (using new NWS formula) • SHOW ALL EQUATIONS USED AND CALCULATIONS