3. First Law of Thermodynamics and Energy Equation

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1 3. First Law of Thermodynamics and Energy Equation 3. The First Law of Thermodynamics for a ontrol Mass Undergoing a ycle The first law for a control mass undergoing a cycle can be written as Q W Q net(cycle) W net(cycle) This statement is experimentally observed by James Joule in 9 th century.

2 3. The First Law of Thermodynamics for a hange in State of a ontrol Mass onsider the following cycle W Q () () () W W Q Q () W W Q Q B B B A B A

3 Q A ( Q Q W) A W A W ( Q W) Because A and is arbitrary processes between state and, we conclude that Q W must be a point function and must be the differential of a property. This property is the energy (E) [J] of the mass: de Q W The property E represents all forms of the energy. It is convenient to separate E in this manner: E = U + KE + PE

4 Internal energy (U) [J]: a property representing all microscopic forms of energy. Kinetic energy (KE) and Potential energy (PE) are in the macro scale. Thus, the first law of thermodynamics for a change of state du d(ke) d(pe) Q W For kinetic energy and potential energy d(ke) d ( mv For potential energy d(pe) d (mgz) ) mg dz m d ( V The first law of thermodynamics is du m d( V ) mg dz ) Q W

5 Integrating the above equation yields (U mg (Z U ) Z ) m ( V Q V W Notes: This law cannot be mathematically proven. Another name of this law is the conservation of energy: two ways to change the energy of a system are heat and work crossing the system boundary. As energy quantities, heat and work are not different. The first law gives only the changes in U, KE, and PE not the absolute values. Since KE and PE are extrinsic properties, U is also an extrinsic property. )

6 3.3 Definition of Work Work (W) [N-m or J]: a force acting through a displacement x W F dx W Sign convention: Work done by a system (+) Work done on a system () F dx Work is a form of energy being transferred across a system boundary.

7 Work associated with a rotating shaft: W F dx F r d T d Power ( W work ) [W], time rate of doing W W FV T or W W t t

8 Specific work (w) [J/kg]: work per unit mass W w m For work SI unit N-m = J other units kwh = 3600 kj For power, SI unit J/s = W other units hp (mechanical) = W hp (metric) = W For specific work, SI unit J/kg = (m/s)

9 3.4 Work Done at the Moving Boundary of a Simple ompressible System onsider a gas in a piston/cylinder device undergoing a quasi-equilibrium process If P is the pressure of the gas, the work done by the system is W W P A dl P dv

10 In case that the relationship between P and V is available as a diagram: We conclude that W W P dv = area under the curve - = area a---b-a

11 However, we can choose different quasi-equilibrium processes from state to state (process A, B or ) Thus, work done during each process is different. As a result, work is not only a function of the initial and end states. work depends on the path of each process. work is a path function. W is an inexact differential.

12 On the other hand, thermodynamics properties are point functions. Notes: symbolic convention For a point function (such as V) the differentials are exact by using symbol dv dv V V For a path function (such as W) the differentials are inexact by using symbol W W W To determine the work from the area under P-V curve, the relationship between P and V must be obtained.

13 The most common relationship between P and V is a polytropic process: PV n onstant PV P V PdV n This result is valid for any values of n except. For n =, PdV P V V ln V which is equivalent to the isothermal process of an ideal gas

14 3.5 General Systems that Involve Work Other types of work are Stretched wire W dl W L Where is tension [N] Surface line W da W A Where is surface tension [N/m] Electric W dz i dt Wire da Liquid Film where is electric potential [V] Z is electric charge [] i is electric current [A] dl

15 i W t W W t i E Battery ombining all types of work gives W P dv dl da dz... W P V V A i... Notes: dz + The process above contributes no work due to nonequilibrium process. (a) Work done on the system (b) No work done on the system

16 3.6 Definition of Heat Heat (Q) [J]: a form of energy which is transferred by the difference of temperature Heat is a path function Q Rate of heat transfer (heat rate) ( Q ) [W]: heat transferred per unit time Q Q or Q Q t t Specific heat transfer (q) [J/kg]: heat per unit mass Q q m Adiabatic process: a process, in which there is no heat transfer Q

17 3.7 Heat Transfer Modes Three modes of heat transfer: onduction: heat transmitted by diffusion of energy through a medium without the bulk motion Fourier's law of conduction dt Q k A dx where k is thermal conductivity [W/m-K] onvection: heat transferred by the motion (or flow) of a medium Newton's law of cooling Q h A (Ts Tsur ) where h is convective heat transfer coefficient [W/m -K] Radiation: heat transmitted by electromagnetic waves in space Stefan-Boltzmann's law Q 4 4 A ( Ts Tsur ) where is emissivity, is Stefan- Boltzmann constant [W/m -K 4 ]

18 Notes: omparison between heat and work Heat and work are both transient phenomena. Systems never possess heat or work Heat and work are boundary phenomena. Both represent energy crossing the boundary of the system. Both heat and work are path function and inexact differential (a) Heat transfers across the boundary. (b) Work transfers across the boundary.

19 3.8 Internal Energy - a Thermodynamic Property U is an extensive property. Thus, we can define Specific internal energy (u) [J/kg]: U u m In general, the value of the specific internal energy must be given relative to a reference state (which is arbitrary). For water, u f = 0 at 0.0 o Note: the values of the specific internal energy can be negative. For the two-phase mixture, u ( u u f f x) u f x (u x u g fg x u u f g )

20 u represents the sum of all microscopic forms of energy. u u u u ext molecule translation int molecule u ext molecule originates from the intermolecular force between molecules. u translation originates from the translation of molecules u int molecule originates from many sources such as molecular rotation, molecular vibration, electron translation, electron spinning, atomic bond, nucleus bond. Sensible energy: all kind to kinetic energies as a function of temperature Latent energy: intermolecular force for phase change process hemical energy: atomic bond Nuclear energy: nucleus bond

21 3.9 Problem Analysis and Solution Technique. What is the control mass or control volume? draw a diagram to illustrate heat/work flows. What do we know about the initial state? 3. What do we know about the final state? 4. What kind of the process takes place? Is anything constant or zero? 5. Is it helpful to draw p-v diagram? 6. What is the thermodynamics model we use (steam tables, ideal gas, etc)? 7. What is our analysis (find work, the first law, the second law, etc)? 8. How do we proceed to find what we need?

22 3.0 The Thermodynamics Property Enthalpy onsider a quasi-equilibrium constantpressure process: P V ) (U ) V P (U P V ) V (P U U V ) P (V U U PdV U U W U U Q U + PV is a property because it is a combination of other properties.

23 Enthalpy (H) [J]: H U PV Specific enthalpy (h) [J/kg]: H h u P v m Thus, for a quasi-equilibrium constantpressure process Q H H Notes: ) Some thermodynamics tables does not contain data of u. Therefore, we can find u from u = h P v. ) Some substance use enthalpy as a reference state instead of internal energy (such as R-, Ammonia). For the two-phase mixture, h ( h h f f x) h f x (h x h g fg x h h f g )

24 3. The onstant-volume and onstant-pressure Specific Heats From the first law: PdV du W du Q. onstant volume ( 0 PdV W ) Specific heat (at a constant volume) ( v ) [J/(kg-K)]: v v v v T u T U m T Q m. onstant pressure ( dh Q ) Specific heat (at a constant pressure) ( p ) [J/(kg-K)]: P P P p T h T H m T Q m v and p are also thermodynamics properties

25 For solid and liquid (incompressible substance: v = constant) v P u and h can be written as dh du du v dp dt If is a constant, we have u h u T dt v dp T T T v P h P In most cases, if P P is not large, we can neglect the second term of the right side. As a result, h h T T for various solid and liquid are listed in Tables A.3 and A.4 Notes: For some substances, could changes significantly at a very high temperature.

26 3. The Internal Energy, Enthalpy, and Specific Heat of Ideal Gases For internal energy, u is generally a function of two independent properties. For a low-density gas u depends strongly on T, but not much on P. Thus for an ideal gas Pv = RT and u = f(t) only For v, we mathematically write u General cases : v T v du Ideal gases : vo dt subscript "o" denotes for the specific heat of an ideal gas

27 Thus, we can write du vo dt u u vo dt For enthalpy, from the definition of h = u + P v, in case of an ideal gas : h h u P v f(t) only u RT For p, we mathematically write h General cases : P T P dh Ideal gases : Po dt dh PodT h h Po dt

28 Because u = f(t) and h = f(t), vo = f(t) and Po = f(t) as well. The values of vo and Po at standard condition are given in Table A.5. The variations of Po as a function of T are given in Table A.6. The main factor causing vo and Po to vary with temperature is molecular vibration.

29 Monoatomic gas: vo and Po are a very weak function of T Diatomic gas: vo and Po are somewhat a function of T Polyatomic gas: vo and Po are strongly a function of T For an ideal gas h dh Po u P v du RdT dt dt Po vo vo on a molar basis Po vo R u RdT R RT To utilize the specific heat, there are three possible ways to find the enthalpy difference of an ideal gas

30 onstant specific heat u u vo ( T T ) h ( T T ) h Po Direct integration: an expression of Po as a function of T (3 rd degree polynomial) in Table A.6 Use Table A.7 and A.8, which are the integration results of the internal energy and the enthalpy over a reference point.

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