AME 60634 Int. Heat Trans. Work Examples [1] Sliding Block Δx F CM work done to the control mass so it is energy gained [2] Shear Work on a Fluid Belt W shear stress × speed × area W˙ = t × v x × A vx t CM Liquid Bath D. B. Go é N m 2ù éJ ù êëm2 s m úû = êë s úû = [W ] work done to the control mass so it is energy gained 1 AME 60634 Int. Heat Trans. Work Examples [3] Boundary Displacement Gas Expansion W p0 boundary work Δz W= CM p1 Vf ò p dV [ J] Vi work done by the control mass so it is energy lost Strain (Compression/Expansion) F boundary work Δz D. B. Go CM1 Vf zf Vi zi (constant area) W = ò s dV = ò s A dz work done to the control mass so it is energy gained [ J] 2 AME 60634 Int. Heat Trans. Work Examples [4] Shaft/Propeller CM torque × angular speed W = T ×q [ W] work done to the control mass so it is energy gained W [5] Electrical Work (Heat Generation) CM Joule (or resistive or Ohmic) heating R W V W = i ×V = = i 2 R R 2 [ W] work done to the control mass so it is energy gained +D. B. Go V 3 AME 60634 Int. Heat Trans. Work Examples [6] Surface Tension CM air Soap bubble surface tension × area change straw CM work done to the control mass so it is energy gained Soap film inside a wire movable wire ΔA D. B. Go 4 AME 60634 Int. Heat Trans. Work Examples [7] Spring Compression F = kx dW = Fdx = kxdx F xf Δx D. B. Go 1 W = ò kx dx = k ( x 2f - xi2 ) 2 xi [ J] 5 AME 60634 Int. Heat Trans. Enthalpy We can literally define a new specific property enthalpy as the summation of the internal energy and the pressure × volume (flow work) h = u + pv ® H =U + pV Porter, 1922 Thus for open systems, the first law is frequently written as æ 1 2 ö æ 1 2 ö dECV = Q -Wnet + å min ç h + vx + gz ÷ - å mout ç h + vx + gz ÷ è 2 øin out è 2 øout dt in D. B. Go 6 AME 60634 Int. Heat Trans. Property, State, and Process • Property is a macroscopic characteristic of the system • State is the condition of the system as described by its properties. • Process changes the state of the system by changing the values of its properties – if a state’s properties are not changing then it is at steady state – a system may undergo a series of processes such that its final and initial state are the same (identical properties) – thermodynamic cycle • Phase refers to whether the matter in the system is vapor, liquid, or solid – a single type of matter can co-exist in two phases (water and steam) – two types of matter can co-exist in a single phase (a water/solvent mixture) • Equilibrium state occurs when the system is in complete mechanical, thermal, phase, and chemical equilibrium  no changes in observable properties D. B. Go 7 AME 60634 Int. Heat Trans. Properties • extensive properties (dependent on size of system) – U internal energy – V volume – S entropy [kJ] [m3] [kJ/K] H enthalpy (total energy) [kJ] m mass [kg] • intensive properties (independent of size of system) – – – –  density T temperature p pressure x quality [kg/m3] [K] [Pa] [-] • specific properties: the values of extensive properties per unit of mass of the system [kg-1] or per unit mole of the system [kmol-1] (inherently intensive properties) – u specific internal energy [kJ/kg] – v specific volume [m3/kg] – s specific entropy [kJ/(kg-K)] D. B. Go h specific enthalpy h = u + pv [kJ/kg] 8 AME 60634 Int. Heat Trans. Pure Substances, Compressible Systems p-v-T Relationship seek a relationship between pressure, specific volume, and temperature • from experiment it is known that temperature and specific volume are independent • can establish pressure as a function of the others p = f (v,T ) p-v-T surface water single phase: all three properties are independent (state fixed by any two) two-phase: properties are dependent on each other (state fixed by specific volume and one other) • occurs during phase changes saturation state: state at which phases begins/ends D. B. Go 9 AME 60634 Int. Heat Trans. Pure Substances, Compressible Systems p-v-T Surface Projections phase diagram • two-phase regions are lines • triple line is a triple point • easily visualize saturation pressure & temperature D. B. Go p-v diagram • constant temperature lines (isotherms) 10 AME 60634 Int. Heat Trans. Pure Substances, Compressible Systems p-v-T Surface Projections T-v diagram • constant pressure lines (isobars) • quality x denotes the ratio of vapor to total mass in two-phase mixture two-phase properties from saturation properties m x= D. B. Go vapor mvapor + mliquid v = (1- x ) v f + xvg 11 AME 60634 Int. Heat Trans. Phase Changes • vaporization/condensation – change from liquid to gas and vice versa • only occurs below critical point • above critical point, the distinction between the two states is not clear • melting/freezing – change from solid to liquid and vice versa • only occurs above triple point • below triple point, the liquid state is not possible and solids change directly to gas (sublimation) D. B. Go 12 AME 60634 Int. Heat Trans. Evaluating Liquid Properties For liquids, specific volume and specific internal energy are approximately only functions of temperature v(T,p) ≈ vf(T) u(T,p) ≈ uf (T) h(T,p) ≈ uf (T)+pvf(T) When the specific volume v varies little with temperature, the substance can be considered incompressible it follows (saturated liquid) æ ¶u ö du cv = ç ÷ = è ¶T øv dT æ ¶h ö du h = u (T ) + pv ® cp = ç ÷ = è ¶T ø p dT cp = cv D. B. Go incompressible thus liquids Changes in u and h can be found by direct integration of specific heats 13 AME 60634 Int. Heat Trans. Compressibility Factor Compressibility Factor universal gas constant R pv Z= RT 8.314 kJ/kmol∙K 1.986 Btu/lbmol∙oR 1545 ft∙lbf/lbmol∙oR R R= M (molecular weight) At states where the pressure p is small relative to the critical pressure pc (where pR is small), the compressibility factor Z is approximately 1. Virial equations of state: D. B. Go Z =1+ B̂ (T ) p + Ĉ (T ) p2 + D̂ (T ) p3 +... 14 AME 60634 Int. Heat Trans. Evaluating Gas Properties At states where the pressure p is small relative to the critical pressure pc (where pR is small), the compressibility factor Z is approximately 1. For ideal gas, specific internal energy and enthalpy are approximately only functions of temperature Specific heat du cv = dT Z =1® pv = RT u(T,p) ≈ u(T) h(T,p) ≈ u(T)+pv = u(T)+RT and ideal gas ≈ h(T) dh cp = dT Changes in u and h can be found by direct integration of specific heats D. B. Go 15 AME 60634 Int. Heat Trans. Heat Transfer • Heat Transfer is the transport of thermal energy due to a temperature difference across a medium(s) – mediums: gas, liquid, solid, liquid-gas, solid-gas, solid-liquid, solid-solid, etc. – Thermal Energy is simply the kinetic energy (i.e. motion) of atoms and molecules in the medium(s) • Atoms/molecules in matter occupy different states – translation, rotation, vibration, electronic – the statistics of these individual molecular-level activities will give us the thermal energy which is approximated by temperature • Heat Transfer, Thermal Energy, and Temperature are DIFFERENT. DO NOT confuse them. • Heat generation (electrical, chemical, nuclear, etc.) are not forms of heat transfer Q but forms of work W D. B. Go – Q is the transfer of heat across the boundary of the system due to a temperature difference 16 AME 60634 Int. Heat Trans. Definitions Quantity Thermal Energy Meaning Symbol/Units Energy associated with molecular behavior of matter U [J] – extensive property u [J/kg] – intensive property Temperature Means of indirectly assessing the amount of thermal energy stored in matter T [K] or [°C] Heat Transfer Thermal energy transport due to a temperature gradient (difference) various Heat Heat Rate/Heat Flow Heat Flux D. B. Go Thermal energy transferred over a time interval (Δt > 0) Q [J] Thermal energy transferred per unit time ˙ [W] q, q˙, Q Thermal energy transferred per unit time per unit surface area q¢¢ [W m2 ] Heat Transfer 17 AME 60634 Int. Heat Trans. Modes of Heat Transfer • Conduction & convection require a temperature difference across a medium (the interactions of atoms/molecules) • Radiation transport can occur across a vacuum D. B. Go 18