Atmospheric Thermodynamics First Law of Thermodynamics A system possesses macroscopic kinetic and potential energy as well as internal energy (u) due to the kinetic and potential energy of its molecules or atoms. q - w = u2 - u1 where q - thermal energy (heat) received by a system (J) w - external work done by the system (J) u1, u2 - internal energy of system before and after change (J) Changes in internal kinetic energy are manifested as changes in temperature. Changes in internal potential energy are caused by changes in the relative positions of molecules due to forces that act between the molecules. This relationship can be expressed in differential form as: dq - dw = du where dq - differential increment of heat added to the system dw - differential increment of work done by the system du - differential increase in internal energy of the system Both of these equations are statements of the first law of thermodynamics.Work done by a gas Consider a gas contained in a cylinder that is fitted with a piston: The volume (V) and pressure (p) exerted by the gas can be shown on a p-V diagram If the gas expands (increase in V) it will push the piston a distance dx and will do an amount of work (dW) given by: dW = Fdx The force exerted by the gas is equal to pA (where A is the cross-sectional area of the piston). dW = pAdx = pdV The work done by the gas (dW) is equal to the pressure exerted by the gas multiplied by the change in volume (dV) of the gas, and is indicated by the shaded area under the curve on the p-V diagram When the gas passes from state A to state B the work done is given by: € dW∫= pdVV1V2∫ When V1 > V2 (the gas is compressed) work is done on the gas and W<0 When V2 > V1 (the gas expands) work is done by the gas and W>0 The work done in going from volume V1 to volume V2 depends on the path of integration and as such is not an exact differential.Dividing by the mass of the gas gives the specific work: € dw∫= pdαα1α2∫ or dw = pdα and the first law of thermodynamics is then expressed as: dq = du + pdα Specific Heats Specific heat: The ratio of the heat added to a system to the change in temperature of the system (=dq/dT) The units for specific heat are J kg-1 K-1 The value of the specific heat depends on how the material (gas) changes as heat is added. Specific heat at constant volume (cv) € cv=dqdT      v const At constant volume a gas does no work and the first law of thermodynamics reduces to dq = du Then cv can be expressed as: € cv=dudT      v const and du = cvdT Using this expression for cv the first law of thermodynamics is: dq = cvdT + pdαInternal energy (u) is a function of state, and changes in internal energy depend only on the initial and final states and not on the path taken between these states. Therefore: € duu1u2∫= u2− u1= cvdTT1T2∫ Specific heat at constant pressure (cp) Defined as: € cp=dqdT      p const In this case work is done by the gas, since as heat is added to the gas the gas expands (dW = pdV). Therefore some of the heat added to the gas goes into doing work, and for a given change in temperature more heat must be added to a gas for a constant pressure process than for a constant volume process, and cp>cv The relationship between cv and cp can be found by taking the first law of thermodynamics (dq = cvdT + pdα) and rewriting it as dq = cvdT + d(pα) - αdp Combining this with the derivative of the ideal gas law (pα = RT): d(pα) = d(RT) = RdT gives dq = cvdT + RdT - αdp = (cv + R)dT - αdpFor a constant pressure process αdp = 0 and € dqdT      p const≡ cp= cv+ R This can be used to express the first law of thermodynamics as dq = cpdT - αdp The value of cp and cv for dry air are: cp = 1004 J kg-1 K-1 cv = 717 J kg-1 K-1 Enthalpy Enthalpy is a function of state defined by: € h ≡ u + pα Since u, p, and α are functions of state enthalpy is also a function of state. Differentiating the definition of h gives: dh = du + d(pα) Using € du = cvdT and dq = cvdT + d(pα) - αdp gives dq = dh - αdp which is another form of the first law of thermodynamicsFrom dq = cpdT - αdp we see that dh = cpdT and in integrated form h = cpT From this equation we note that h corresponds to the heat required to raise the temperature of a material from 0 to T K at constant pressure. When heat is added to air at constant pressure dq = dh = cpdT and the enthalpy increases. Enthalpy is also commonly referred to as sensible heat. Special Processes Referring to the first law of thermodynamics expressed as: dq = cvdT + pdα or dq = cpdT - αdp we can consider a number of special processes. Isobaric process: dp = 0 € dq = cpdT =cpcv      cvdT =cpcv      du Isothermal process: dT = 0 € dq = −αdp = pdα= dw Isochoric process: dα = 0 € dq = cvdT = duAdiabatic process: dq = 0 € cpdT =αdpcvdT = − pdα An adiabatic process is one in which a material undergoes a change in its physical state (e.g. pressure, volume, or temperature) without any heat being added to it or withdrawn from it. Consider isothermal and adiabatic compression illustrated on a p-V diagram: For both processes V and α decrease For the isothermal process (shown by curve AB) this implies that p must increase For the adiabatic process (shown by curve AC) the internal energy, and thus temperature, increases. For the same mass of gas at the same volume (points B and C) the sample with the higher temperature (C) will also have a higher pressure, hence the adiabat (AC) on the p-V diagram is steeper than the isotherm (AB)Air parcel – a small mass of air that is thermally insulated from its environment, such that its temperature changes adiabatically, and that has exactly the same pressure as its environment Dry Adiabatic Lapse Rate Consider an air parcel undergoing an adiabatic change in pressure, with no phase change of any water substance in the air parcel. For this air parcel the first law of thermodynamics can be written as: € cpdT =αdpcpdT =RTpdp Since this change in pressure implies a change in elevation: € cpdTdz=RTpdpdz From the hydrostatic equation: € dpdz= −ρg Combining these equations give: € dTdz= −ρgRTcpp= −gcp= −−9.81 …