A/2 l,k. Problem 1 STRATEGY. KNOWN Resistance of a complete spherical shell: r rk. Inner and outer radii

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1 Prblem 1 STRATEGY KNOWN Resstance f a cmplete sphercal shell: R ( r r / (4 π r rk sphere Inner an uter ra r an r, SOLUTION Part 1: Resstance f a hemsphercal shell: T calculate the resstance f the hemsphere, use analgy wth slab: T A l,k The heat flux thrugh a slab s: T q '' R T l ka Nw, fr the slab f same length, l, half the area (A/ an the same heat flux: T A/ l,k T T T q '' l R' R A k Therefre, fr half the area, the resstance s uble R' R Thus, the resstance f the hemsphere twce f that f the cmplete sphere: R r r π r rk hem sphere Rsphere (Ans.

2 Part : Freezng tme fr hemsphercal tssue regn: Parameters assume: T m Freezng temperature ( C T Cl ar temperatur e (C latent heat f fusn (J/kg H f h he ρ at transfer ceffcent (W/m K 3 ensty (kg/m k thermal cnuctvty f the frzen layer (W/ m K Nw, heat lss thrugh a frzen layer at a stance r frm the center (see Fgure fr a cnvectn bunary cntn: T T T T q R Ah r r rrk r h (1 m m hemsphere + 1/ ( /( π + 1/ π Where, A π s the uter surface area f the hemsphere r Energy rate frm the latent heat gven ff at the freezng frnt f thckness r (Nte that r s ecreasng s r s negatve r t f π ρ ( q H r Equatn (1 an ( frm abve an slvng fr t, tme t freeze as a functn f r: Tm T r H f πr ρ r r 1 + t πrrk πrh Tm T r r 1 t + π H fρ πrrk πrh Tm T 1 t r r rr + H fρ rk rh t r f r r r r r r m Tm T r r 1 t rr + H ρ rk r h r r r

3 r r Tm T tm 1 r r [ ] r H fρ k r r r r h t r + r r r r 1 r + k 3r rh 3 r r r Tm T t 1 [ ] m 1 r r r r r r t + H fρ k 3r rh r r r r 1 r + + r k 3 3r 3h r r r r + r r k 6 3r 3h r t m 3 3 H f ρ 1 r r r 1 r + + r Tm T k 6 3r 3h r (Ans. (3 Part 3: Checkng fr cnsstency wth sphere frmula In (3 abve, fr r an h, the equatn reuces t: t m H f ρ r kt ( m T 6 ( Ans. (4 whch s cnsstent wth the frmula fr freezng a sphere f raus r.

4 Prblem STRATEGY KNOWN Surface temperature f humans T 33 C 36 K Infrare thermgraphy perates n the wavelength range 6 14 µ m SOLUTION Maxmum energy emtte by the human surface (cverng the entre wavelength range: '' 4 qmax σt (1 where, 8 4 σ W m K T 36 K Substtutng n (1 abve, '' q W m ( max Nw, fractn f energy between ( λ 14 µ m an ( λ 1 6 µ m Fλ T FλT (3 λ T 1836; F λ T 484; F Thus, Fnet F 484 F (4 Therefre, fractn f energy emtte by the human surface n the wavelength range 6 14 µ m q '' Fnetq '' max 41.1 W m (Ans. (5

5 Prblem 3 STRATEGY KNOWN There are three raatng surfaces the human by surface, the heate plate an the rest f the surrunngs. Emssvty f clthe by s 1 ASSUMPTIONS 1 Heate surface an rest f surrunngs apprxmate as a black by Persn s assume as vertcal cylner SOLUTION Part 1: Net raatve exchange between the persn (1 an the heate surface ( 4 4 q1 σ A1 F1( T p T h (1 Where, A1 surface area f the persn (m F1 Vew Factr frm persn (1 t heate surface ( Tp persn surface temperature ( K Th heate surface temperature (K 8 4 σ Stefans-Bltzman Cnstant ( W/m K Part : Relatnshp between sze x f the heate surface an ar velcty ver the persn s surface Fr the persn: IN OUT + GENERATION STORAGE ( IN: OUT: nne σaf ( T T σaf ( T T ha ( T - T 1 1 p h 1 13 p a 1 p b GENERATION: Qs A 1

6 STORAGE: nne (steay state Pluggng n ( abve σaf ( T T σ AF ( T T ha ( T - T + Q A (3 1 1 p h 1 13 p a 1 p b s 1 IN OUT GEN. Nw, nfrmatn n ar velcty u s embee n the heat transfer ceffcent, h. Nu D hd n BRe D Pr k 1/3 ρu D B µ Frm abve, n u h n 1 µ D 1/3 BPr ρd k n Pr 1/3 (4a (4b The nfrmatn n length, x s present n the vew factr frm the persn t the heate plate, F1. Wrtng h as a functn f F1: σa1 F1( Tp Th σa1 F13( T p Ta + Q s A1 ha1 ( Tp - Tb (5 Substtutng fr h n (6 frm (4b an rearrangng: Q sf1( Tp Th σ F13( Tp Ta h (6 ( T - T p b u n n Q sf1( Tp Th σ F13( Tp Ta 1 µ D 1/3 ( Tp - Tb BPr ρd k (Ans. Part 3: Cnvectve heat lss frm the by In rer t btan the cnvectve heat lss frm the by, we frst nee t btan the cnvectve heat transfer ceffcent, h. The persn assume as a vertcal cylner, therefre, the fllwng crrelatn t calculate h hls g (fr bth lamnar an turbulent flws: n 1/3 Nu BRe Pr (7 D D

7 Nte, prpertes f ar n abve Eqn. (7 nee t be btane at the flm temperature, Tf GIVEN PARAMETERS D.45 m v.1 m/s Tp 33 C 36 K T C 73 K b Thus, T 16.5 C 89.5 K f Prpertes f ar at Tf : µ ρ kg/m Pa s 3 k.547 W/m K Pr.71 T f Tp +Tb (8 ρvd Reynls N., ReD µ Flw s Lamnar. Therefre, frm table, B.683 an n.466 hd.466 1/3 NuD.683(Re D ( Pr k k.466 1/ /3 h.683(re D ( Pr.683 (346.8 (.71 D.45 h 1.45 W/m K Cnvectve lss, q'' cnv ht ( p Th 1.45( W/m (Ans. Part 4: Length, x f the heate surface T calculate the length, x, we nee t frst fn the vew Factr F1. Then, use the chart that relates F1 wth X an Y t btan the length, x. Frm Eqn. (3 abve:

8 σaf ( T T + σ AF ( T T + ha ( T - T Q A (9 1 1 p h 1 13 p a 1 p b s 1 σf T T + σf T p T a + ht T s ( ( p h 13( ( p - b Q Usng the fllwng relatnshp an substtutng fr F13 n Eqn. (9 abve: F F + F F p h 1 p a p b s (11 σf ( T T + σ (1 F ( T T + ht ( - T Q (1 Rearrangng Eqn. (11 t btan an explct expressn fr F1 4 4 Q sht ( p - Tb σ ( Tp Ta σ ( Ta Th F (13 Nte, n Eqn. (1, q'' ht ( T s the cnvectve lss calculate n Part 3 abve cnv p b All terms n the R.H.S f Eqn. (13 are knwn: Q s 7 W/m Tp 33 C 36 K Tb C 73 K T C 473 K h q '' cnv W/m (frm abve Pluggng n numbers an slvng fr F1 W W W (36 73 K Q 4 s q '' cnv σ ( Tp T m m m K a σ ( T 8 W a Th ( K 4 F m K Thus, F 1.63 (14 In rer t rea the chart we nee t multply F1 wth a factr f F Frm the chart, fr 18.5F an X Y, the value f X crrespns t 1.8 (see chart belw

9 x X 1.8 an z.5 m z x 1.8z m (Ans

10 Prblem 4 STRATEGY ASSUMPTION Assume fsh has a mass M g KNOWN t 84 ays c μg/g α.8 E.7 1/ay c cnc. at any nstant Dlutn -G.c K I /ay.8 1/ ay G.7 1/ay c.13 μg/g OUT (E, K IN (I Part 1: Mass Balance fr mercury n the fsh ver tme Δt IN OUT + GENERATION STORAGE (1 IN Mα Ic OUT ME ( + Kc GEN. MGc STORAGE M c Here, GENERTION refers t the lutn f mercury cncentratn ue t fsh grwth Substtutng n (1 abve, [ ] MαIc M ( E + K c MGc t M c (Ans. ALTERNATIVE METHOD:

11 Snce the fsh grws, ts mass changes. S, the STORAGE term can be alternatvely wrtten as: STORAGE ( cm ( c M + M c The specfc grwth rate, G, s the change n by weght per by weght per tme.e. M G (3 M t Substtutng fr ΔM frm abve: M GM t (4 STORAGE cgm + t M c (5 Nte, n ths apprach there s n GENERATION term; IN an OUT reman the same as abve: IN OUT STORAGE MαIc t M ( E + K c t M c + cmg t (6 Rearrangng Eqn.(6, we get the same expressn as abve [ ] MαIc M ( E + K c MGc t M c (Ans. Part : Dfferental equatn fr the cncentratn f mercury,c as a functn f tme, t [ ( ] Mα Ic M E + K c MGc t M c c α Ic ( E + K + G c t c α Ic ( E + K + G c (Ans. t Part 3: Slve the Dfferental equatn fr the cncentratn f mercury,c as a functn f tme, t Let, A α Ic an B E + K + G Therefre, the fferental equatn n terms f A an B: c A Bc t (7 c A Bc t Integratng bth ses t slve fr c(t:

12 ct ( c c A Bc t t Pluggng n the values f A an B: 1 [ ( ] ct ( ln A Bc c t B 1 A Bc( t ln t B A Bc A Bc t A Bc ct ( ( Bt e A ( A Bc e B Bt αic [ αic ( E + K + G c ] e E+ K + G ( E+ K+ Gt ( (A ct ns. Part 4: Mercury cncentratn nse fsh after 84 ays Frm abve, αic [ ( ] e αic E + K + G c ct ( E+ K + G ( E+ K+ Gt All values n the R.H.S are knwn. Fr n ntal cnc., c, Eqn. (8 reuces t: α Ic ( 1 ct ( E+ K + G ( E+ K+ Gt e Pluggng n the values f α, EKGIc,,,, n Eqn. (9 an slvng fr tme t 84 ays: ay 1 μg + + ay ay ay ( e ay g ct ( ay ay ay (8 (9 (1 Therefre, ct ( 84 ays.41 μg/g (Ans.