6 - Unsteady State Mass and Energy Balances_S10

6 - Unsteady State Mass and Energy Balances_S10 - Unsteady...

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Copyright 2010 D.B. Dadyburjor and J. A. Shaeiwitz Unsteady State Mass and Energy Balances ChE 202 Spring 2010
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Copyright 2010 D.B. Dadyburjor and J. A. Shaeiwitz Retrospective – Sub-processes considered 9 Change in pressure at constant temperature 9 Change in temperature at constant pressure 9 Change in T or P at constant volume 9 Change in phase at constant P and T 9 Mixing of pure components to multicomponent system at constant T and P (and reverse) 9 Pure components in stoichiometric amounts reacting to form pure products(s) 9 Combine to deal with more-complicated processes; 9 Use algebraic MB, EB balance equations -- Difference equations for closed systems (no input/output terms) (v sys 2 )/2 + g z sys + = Q (in) /m sys –W s (by) /m sys –P ext Steady-state equations for open systems (no accumulation term) -- to solve for needed quantities ) ( ) ( , , , , ) ˆ ˆ ˆ ( ) ˆ ˆ ˆ ( 0 by s in out i i p i k i in i i p i k i W Q m E E H m E E H & & & & + + + + + = sys U ˆ sys V ˆ
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Copyright 2010 D.B. Dadyburjor and J. A. Shaeiwitz Now . .. Use differential equations to solve unsteady-state MB, EB equations (accumulation + input/output terms) ) ( ) ( ) ( , , , , , , ) ˆ ˆ ˆ ( ) ˆ ˆ ˆ ( }) ˆ ˆ ˆ { ( by ext by s in out i i p i k i in i i p i k i sys p sys k sys sys W W Q m E E H m E E H dt E E U m d & & & & & + + + + + = + + formation out i in i sys m m m dt m d & & & + = ) ( Start with Material balances only
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Copyright 2010 D.B. Dadyburjor and J. A. Shaeiwitz Unsteady-State Mass Balances Different ways to write terms: ACCUMULATION IN/OUT FORMATION TOTAL d[ mass ]/dt , d[ moles ]/dt in / out , in / out --- MATERIAL d[ ρ V ]/dt , d[ C tot V ]/dt ( ρ v) in / out , (C tot v) in / out --- (V –system volume) (v –stream volumetric flow rate) COMPONENT d[ m A ]/ dt , d[ n A ]/dt ( A ) in / out , ( A ) in / out (r A )(V) A d[ C A V ]/dt (C A,in v) in / out d[ m tot w A ]/dt , d[ n tot y A ]/dt ( w A ) in / out , ( y A ) in / out
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Copyright 2010 D.B. Dadyburjor and J. A. Shaeiwitz Unsteady-state MB Example – Draining a tank Batch process – unsteady state () h A K dt dh A Kh dt h A d K h K m h m m h A m tank System m dt dm t h tank tank tank out out out tank tank out system ρ = = = = = + = constant are and If constant a is . Assume ) ( for ip relationsh Need 0 0 ) ( for expression an Want & & & & m tank out
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Copyright 2010 D.B. Dadyburjor and J. A. Shaeiwitz Time out – 1: Solving ODEs dy/dt = f(y,t); IC y = y 0 at x = x o ± Want y(t) ± SEPARATION OF VARIABLES ± If f(y,t) consists of separate functions of y & t f(y,t) = p(y) . q(t) dy/dt = p(y) q(t) ± So dy / p(y) = q(t) dt Integrating, P(y) = Q(t) + C C – integration constant ± Use IC to get C : C = P(y o ) – Q(t o ) ± P(y) = Q(t) + P(y o ) – Q(t o ) This yields y(t)
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Copyright 2010 D.B. Dadyburjor and J. A. Shaeiwitz Time out – 2: Solving ODEs dy/dt = f(y,t); IC y = y 0 at x = x o ± Want y(t) ± INTEGRATING FACTOR ± If first-order, linear ODE ± f(y,t) = Q(t) + y P(t) ± Multiply both sides of ODE ( {dy/dt} and {f(y,t)} ) by Integrating Factor (IF) exp [ -P(t).dt] ± Then: y exp[ -P(t)dt]= { Q(t) exp[ -P(t’)dt’]} dt+ C ± Use IC to get C , as before
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Copyright 2010 D.B. Dadyburjor and J. A. Shaeiwitz
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This note was uploaded on 05/14/2010 for the course CHE 102 taught by Professor Dadyburjor during the Spring '10 term at WVU.

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6 - Unsteady State Mass and Energy Balances_S10 - Unsteady...

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