hermodynamics Lecture 7: Heat transfer Open systems Bendiks Jan Boersma hijs Vlugt heo Woudstra March, 00 Energy echnology
Summary lecture 6 Poisson relation efficiency of a two-stroke IC engine (Otto cycle) gas turbine with a heat exchanger open systems mass and energy balance st law of thermodynamics stationary open systems turbine, throttle, heat exchanger March, 00
Heat exchanger st law for stationary open system: ( ) ( ) ( ) W cv + m h h + V V + g z z with: W 0, V V << h h and g z z << h h system I system II I ( ) m h h II ( ) cv ( ) m h h 3 4 3 heat transfer only through wall I /II ( ) ( ) m h h m h h 3 m 3 + 3 4 3 0 II I II m I m m 3 4 March, 00 3
Wind turbine V C V f V 3 4 V f < V 4 f V rea at 4: C / f hrust (F Δp ): (p p 3 ) ½ρV ( f ) Momentum. change (F mδv): ρv C( f ) Balance of forces: C ½(+ f ) Power: W. ΔE. ¼ρ V 3 (+f )( f 8ρV ) 3 W f /3 March, 00 4 max 7
5 ρr E f ( ρ, R, t) E c ( c.033) t kg m α 3 β γ ( kg m ) ( m ) ( s ) α β 5 γ s t 0.03 s, R 30 m E ~ 50 0 J (50 J) March, 00 5
Problem 4.59 Figure P4.44 shows a simple vapor power plant operating at steady state with water circulating through the components. Relevant data at key locations are given on the figure. he mass flow rate of the water is 30 kg/s. Kinetic and potential energy effects are negligible. Determine: a. he thermal efficiency b. he mass flow rate of the cooling water passing through the condenser, in kg/s March, 00 6 η 0.33; Φ m 4485 kg/s
Energy transfer by heat Q E W Energy equation closed system: March, 00 7 E W. Heat flux Q through area : q d. where q is the specific heat flux wo principal ways of heat transfer: conduction and radiation occur due to temperature differences: in general no equilibrium (hermodynamic) equilibrium only for ideal case of quasistatic processes: time does not play a role infinitesimal temperature differences
Heat transfer by conduction Fourier s Law: q x d κ dx where κ is the heat conduction coefficient otal heat flux: d x κ dx where is the surface area (x). q(x) x March, 00 8
Heat transfer by radiation Stefan-Boltzmann Law: q q e e σ 4 b εσ 4 b black body gray body b. qe with: σ 5.67 0 8 Wm K 4 0 ε March, 00 9
distance 50 0 6 km Sun s 5.8 0 3 K R 0.7 0 6 km q σ ( 3) 4 6 5.8 0 64 0 4 8 5.67 0 W/m q πr q 4 ( R ) q 4 π R q q R 3 ( 0.7 50).4 0 6 64 0 W/m 4 ( 0.5q ) 80 K ( + 7 C) σ projected surface total surface 4π R March, 00 0 πr 4 http://www.junkscience.com/gm/
Heat transfer by convection In many practical situations we also have to deal with heat transfer by convection. For this case we have Newton s cooling law (emperical): ( ) h b f where h is the heat transfer coefficient f moving fluid (gas or liquid) solid Q. b system boundary March, 00
Stationary -D heat conduction Stationary: temperature does not change as a function of time; -dimensional: only one independent coordinate (x) (plane wall), (r) (sphere) Plane wall:.. stationary energy equation (E 0, W 0): Q ( x) Q ( x + Δx) or Q ( x) constant ( q x k d dx) with Fourier s law : L 0 neglecting the temperature dependence of k: k ( ) L L k R with thermal resistance R L / k. dx k d March, 00 x L.. (x) x+δx. Q Q(x) Q(x+Δx) R L / k
Multiple layers L L B Q.. Q. Q 3 k k B 3 R L k R B LB k B eliminate : L k 3 k March, 00 3 L 3 3 L k + L k R + R B B B B B
. Q h L. 3 4 Q. L 3 Q h k h 34 h k 34 4 Constant heat flux (with Newton s cooling law, Fourier s law): 3 h ( ) h34 ( 3 4 ) L k Eliminate and 3 : 4 U h + L k + h34 where U is the heat transfer coefficient: U ( ) March, 00 4 h 4 + L k + h 34
Cylindrical walls.. Stationary energy equation (E 0, W 0): Qr ( ) Qr ( +Δr) constant r r ( k kπrl ) d d with Fourier s law : dr r π kl ln π L kd neglecting the temperature dependence of k: Q with thermal resistance ( r r ) R dx R ln dr ( r r ) π kl r r r Δr March, 00 5
Critical isolation thickness with: ( r r) heat transfer rate: i f ln u i R + πkl h πr L ( r r) ln u i + π L k h r u u cr minimum heat resistance (dr / dr u 0): r R u r u u k h u March, 00 6 f.3...0 h u r i k h u r u i r i ( r) ln + r 0.9 0.8 3 4 5 6 7 8 9 0 r u
Exercise thick-walled pipe of stainless steel (k 9 W/m K) with a cm inner diameter and a 4 cm outer diameter is covered by a 3 cm thick insulation layer (k 0. W/m K). he temperature is 600 ºC on the inside and 00 ºC on the outside. Determine: a) the heat transfer rate Q per unit pipe length in W/m; b) the temperature at the pipe/insulation interface in ºC. (nswers: a) 680 W/m, b) 596 ºC ) March, 00 7
Exercise (homework) Water at 50 ºC flows through a pipe with an inner diameter of.5 cm, with an inner heat transfer coefficient of h i 3500 W/m K. he pipe has a 0.8 mm thick wall and a heat conduction coefficient k of 6 W/m K. he heat transfer to the surroundings (air at 0ºC) on the outside of the pipe is by free convection with a heat transfer coefficient of h u 7.6 W/m K. Determine: a) the coefficient U per unit pipe length in W/m K; b) the total heat flux Q per unit pipe length in W/m. March, 00 8 (nswers: a) 0.633 W/m K, b) 9 W/m )
Instructions for study Ch. - 4 of Moran & Shapiro have been discussed; also study now.4. (heat transfer). Study 4.-4.3, including all examples. Study examples in 4.4 on instationary systems. For the exams you need to be able to work with the stationary energy balance. Do exercises 4.0, 4. & 4.3. For 4., use mass and energy balances for multiple inlets and outlets We have discussed from Mills Basic Heat & Mass ransfer ( nd Ed.):.3., first part of.3. (radiation), Ch. till.3.3. Heat transfer by radiation (i.e., h c,o ) will not be required for the exam Do the exercises on heat transfer (homework) March, 00 9