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# Problems Ener Rad ATSC 5003

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This 16 page Class Notes was uploaded by Tony Bode on Wednesday October 28, 2015. The Class Notes belongs to ATSC 5003 at University of Wyoming taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/230351/atsc-5003-university-of-wyoming in Atmospheric Science at University of Wyoming.

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Date Created: 10/28/15

1 Properties afan Air Parcel Temperature Pressure and Speci c Volume One ofthe things you will discover in ATSC5003 is that dry air has two 2 thermodynamic degrees of freedom In other words measured values of temperature and pressure x the thermodynamic state of dry air Since atmospheric scientists measure the temperature and pressure twice a day at various locations across the globe and at standardized altitudes we know the worldwide state of the air The concept of two thermodynamic degrees of freedom also means that we can derive a few additional properties of the air For example by employing the ideal gas equation of state we can derive the speci c volume symbolized as lower case vee v v Rd T P 1 Here Rd is the speci c gas constant for dry air it has a value of28705 Pa m3 K391 kg39l Show Figure 113 from Iribarne and Godson 30 u unsmmm39 rumian NMquot 5 Fig 1H Thurmouymumc mrilm rm mm gum and pmmmm m 7 y p rum 1 Tmmm In the rst exercise we will use the IDL procedure called pivsiviisothermspro to draw lines of constant temperature in the P v plane We will call these lines isotherms and we will consider the temperatures T 225 K T 300 K and T 375 K Summarizing Given a constant temperature T and an array of pressure P we will calculate the corresponding array of speci c volume v formulated as vRdTP This style of presentation 7 ieP versus v and the related concept of the isothermal expansion is a review of undergraduate physics and chemistry 12105 225 300 375 7 80104 7 7 Pressure Pa 40104 7 05 10 15 20 25 30 Speci c Volume m3 kgquot utmdw 21th NEW 0 dlun 39 Mm htthWW Wan PM quot 1bstevek W quot mhtm 2 Ascent 0r Descent of an Air Parcel The Isem ropic Process Now we consider how the properties of an air parcel change as it ascends For this we will use the IDL procedure called pivsiviisothermsiandiisentropespro First the basic concepts Ascent of an air parcel implies motion to higher altitude where the pressure of the atmosphere 7 commonly referred to as the surroundings is lower If mechanical equilibrium is maintained between the air parcel and its surroundings the parcels pressure will conform to the lower ambient pressure present at higher altitude Furthermore we also expect the speci c volume of the parcel to increase ie our experience tells us that the air parcel volume should increase with ascent so it follows that the air parcels volume divided by its mass should also increase with ascent Based on these basic concepts the following is concluded Parcel Ascent gt Decreased Surrounding and Parcel Pressure gt Parcel Expansion Conversely Parcel Descent gt Increased Surrounding and Parcel Pressure gt Parcel Compression We are interested in the thermal consequence of ascent Using only the ideal gas equation of state T Rd P v we cannot say what happens to the air parcels temperature That is since our experience tells us that pressure decreases and that specific volume increases we cannot say what the consequence is for temperature For example it may be that the decrease in pressure exactly compensates the increase in specific volume with the result being that an air parcels temperature is constant for all altitudes This uncertainty motivates a detailed analysis of thermodynamics In differential form we have an equation describing the constraints st nd 39 1mposed by the the l and 2 laws of thermodynam1cs Tds vdPcpddT 3 Here s is the specific entropy ofthe air parcel and de is the isobaric specific heat capacity of dry air 1005 J K391 kg39l Substituting the ideal gas equation of state for specific volume v Rd T P separating variables and integrating from a state 1 to a state 2 we have P T s2 s1 Rd 1n172 cpd lnT 2 4 l 1 This equation is important because to rst approximation speci c entropy does not change when a parcel ascends or descends If this approximation is valid the left hand side of equation 6 is zero and states 1 and 2 are said to be connected by a constant speci c entropy path Commonly this path is referred to as an isentropic path or as an isentrope In Atmospheric Science you will also hear reference to an adiabat which is nearly the same thing as an isentrope Enough about the semantics here is how the isentropic path is described mathematically P2 T2 0 Rd de Equation 5 and algebra can be used to derive equation 6 RdC d P P 2 6 54 We will use equation 6 to derive the temperature T2 corresponding to a pressure P2 For this we also need to specify an array of pressure which will call P2 and an initial state of the dry air speci ed by measurements of T1 and P1 Summarizing Using Equation 6 a P2 corresponding to various altitudes measurements of temperature and pressure at an initial state T1 and P1 and the ideal gas equation of state we can derive T2 as R C P2 d pd st nd 39 39 39 39 combined 1 and 2 laws of thermodynamics dry 1sentrop1c process is assumed T 2 T1 and we can derive the corresponding v2 as v2 Rd T2 P2 ideal gas equation of state for dry air The plot of P2 versus v2 is shown on the next page solid lines Because there are three initial states ie T1 225 K and P1 10x105 Pa T1 300 K and P1 10x105 Pa T1 375 K and P1 10x105 Pa three 39 r are quot J Also quot39 J are the lines of constant temperature isotherms dotted lines discussed in the previous section As an example isentropic ascent from T1 300 K and P1 10XlO5 to P2 035X105 Pa produces atemperature T2 225 K ie 75 degrees of cooling 12105 80104 Pressure Pa 40104 05 10 15 20 25 30 Speci c Volume m3 kgquot We conclude the following about isentropic ascent Isentropic Ascent gt Expansion gt Cooling For isentropic descent Isentropic Descent gt Compression gt Warming 3 Ascent 0r Descent of an Air Parcel The Isem ropic Process Now we are going to repeat the previous problem This time we are going to de ne the speci c gas constant R d and the isobaric speci c heat capacity C pd as global constants The other thing that will be different is that we are going to write a function that calculates the speci c volume Results are shown in the following IDL procedure quotpivsiviisothermsiandiisentropesiuniviconstantspro You should note the following 1 R d is available to the function as a global The name of the global is Rd 2 There is no need to repeat the de nition of Rd within the function 3 By using the global there is no need to hardwire ie R d appears as Rd in the formulas seen in both the main procedure and in the function In your programming try to avoid hardwiring 4 It is recommended that frequently used constants be de ned as globals 1 Properties afan Air Parcel Temperature Pressure and Speci c Volume One ofthe things you will discover in ATSC5003 is that dry air has two 2 thermodynamic degrees of freedom In other words measured values of temperature and pressure x the thermodynamic state of dry air Since atmospheric scientists measure the temperature and pressure twice a day at various locations across the globe and at standardized altitudes we know the worldwide state of the air The concept of two thermodynamic degrees of freedom also means that we can derive a few additional properties of the air For example by employing the ideal gas equation of state we can derive the speci c volume symbolized as lower case vee v v Rd T P 1 Here Rd is the speci c gas constant for dry air it has a value of 28705 Pa m3 K391 kg39l Figure 113 from Iribarne and Godson JD Arwxsvhtmr im anm39NAmm Flg Hr Thermodymnm wrl39dxc or Mimi gum and pruu LHivHCUILv 117 rum 1 Tf lLlllL Pressure Pa In the rst exercise we will use the IDL procedure called pivsiviisothermspro to draw lines of constant temperature in the P 7 v plane We will call these lines isotherms and we will consider the temperatures T 225 K T 300 K and T 375 K Summarizing Given a constant temperature T and an array of pressure P we will calculate the corresponding array ofspeci c volume v formulated as vRd TP This style of presentation 7 ieP versus v and the related concept of the isothermal expansion is a review of undergraduate physics and chemistry 30 hmsmtku rHHwnm NAMquot s 12105 I I I 225 3q0 37 80104 40104 10 15 20 25 Specific Volume m3 kg391 30 Flg 1H Thcmmdymnm 5mm mum guxm Anti prmmmm y 1 mm 1 rpmm h waLhErlIw 172qu ErairSmlndin 1mm www2tmnslcl2erllIhStevensPi resi rehtml 2 Ascent 0r Descent of an Air Parcel The Isentropic Process Now we consider how the properties of an air parcel change as it ascends For this we will use the IDL procedure called pivsiviisothermsiandiisentropespro First the basic concepts Ascent of an air parcel implies motion to higher altitude where the pressure of the atmosphere 7 commonly referred to as the surroundings is lower If mechanical equilibrium is maintained between the air parcel and its surroundings the parcels pressure will conform to the lower ambient pressure present at higher altitude Furthermore we also expect the speci c volume of the parcel to increase ie our experience tells us that the air parcel volume should increase with ascent so it follows that the air parcels volume divided by its mass should also increase with ascent Based on these basic concepts the following is concluded Parcel Ascent gt Decreased Surrounding and Parcel Pressure gt Parcel Expansion Conversely Parcel Descent gt Increased Surrounding and Parcel Pressure gt Parcel Compression We are interested in the thermal consequence of ascent Using only the ideal gas equation of state T Rd P v we cannot say what happens to the air parcels temperature That is since our experience tells us that pressure decreases and that specific volume increases we cannot say what the consequence is for temperature For example it may be that the decrease in pressure exactly compensates the increase in specific volume with the result being that an air parcels temperature is constant for all altitudes This uncertainty motivates a detailed analysis of thermodynamics In differential form we have an equation describing the constraints imposed by the the lSt and 2quotd laws of thermodynamics Tds vdPcpddT 3 Here s is the specific entropy ofthe air parcel and de is the isobaric specific heat capacity of dry air 1005 J K391 kg39l Substituting the ideal gas equation of state for specific volume v Rd T P separating variables and integrating from a state 1 to a state 2 we have P T s2 s1 Rd ln2 cpd ln2 4 l 1 This equation is important because to rst approximation speci c entropy does not change when a parcel ascends or descends If this approximation is valid the left hand side of equation 6 is zero and states 1 and 2 are said to be connected by a constant speci c entropy path Commonly this path is referred to as an isentropic path or as an isentrope In Atmospheric Science you will also hear reference to an adiabat which is nearly the same thing as an isentrope Enough about the semantics here is how the isentropic path is described mathematically P T 0 Rd ln cpd ln 5 Equation 5 and algebra can be used to derive equation 6 RdC d P P 2 6 TT 21L1 We will use equation 6 to derive the temperature T2 corresponding to a pressure P2 For this we also need to specify an array of pressure which will call P2 and an initial state of the dry air speci ed by measurements of T1 and Summarizing Using Equation 6 a P2 corresponding to various altitudes measurements of temperature and pressure at an initial state T1 and PI and the ideal gas equation of state we can derive T2 as P Rd C pd T2 T1 combined lSt and 2quotd laws of thermodynamics dry isentropic process is assumed l and we can derive the corresponding v2 as v2 Rd T2 P2 ideal gas equation of state for dry air The plot ofPZ versus v2 is shown on the next page solid lines Because there are three initial states ie T1 225 K and P1 10x105 Pa T1 300 K and P1 10x105 Pa T1 375 K and P1 10x105 Pa three isentropes are illustrated Also illustrated are the lines of constant temperature isotherms dotted lines discussed in the previous section As an example isentropic ascent from T1 300 K and PI 10XlO5 to P2 035X105 Pa produces atemperature T2 225 K ie 75 degrees of cooling 120105 800104 Pressure Pa 400104 O I I I I I I I I I I I I I I I I I I I 05 10 15 20 25 30 Specific Volume m3 kg391 We conclude the following about isentropic ascent Isentropic Ascent gt Expansion gt Cooling For isentropic descent Isentropic Descent gt Compression gt Warming 3 Ascent 0r Descent of an Air Parcel The Isentropic Process Now we are going to repeat the previous problem This time we are going to define the specific gas constant Rd and the isobaric speci c heat capacity C pd as global constants The other thing that will be different is that we are going to write a function that calculates the specific volume Results are shown in the following IDL procedure pivsiviisothermsiandiisentropesiuniviconstantspro You should note the following 1 Rd is available to the function as a global The name of the global is Rd 2 There is no need to repeat the definition of Rd within the function 3 By using the global there is no need to hardwire ie Rd appears as Rd in the formulas seen in both the main procedure and in the function In your programming try to avoid hardwiring 4 It is recommended that frequently used constants be defined as globals

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