上海交通大学：《生物传热学教与学 Bioheat Transfer》课程教学资源（电子讲义）lecture 8 Heat Transfer to Blood Vessels

Heat Transfer to Blood Vesses

Heat Transfer to Blood Heat Transfer to Blood Heat Transfer to Blood Heat Transfer to Blood Vessels Vessels Vessels Vessels

Blood Flow Convection Single blood vessel embedded in a tissue cylinder with its outer surface at constant T. Known: 6,P6,C6,k6,T(0),Tm(0) P,C,,6,7, Find: Z(r Ave.(bulk)blood temp. Z(x) Vessel wall temp. Assumptions:steady-state;=O fully developed flow and thermal fields within the vessel->neglecting end effects up-ave.blood flow velocity

Single blood vessel embedded in a tissue cylinder with its Single blood vessel embedded in a tissue cylinder with its Single blood vessel embedded in a tissue cylinder with its Single blood vessel embedded in a tissue cylinder with its outer surface at constant T outer surface at constant T outer surface at constant T outer surface at constant To . Known: Find: Assumptions: steady- state; Assumptions: steady- state; Assumptions: steady- state; Assumptions: steady- state; fully developed flow and thermal fields within fully developed flow and thermal fields within fully developed flow and thermal fields within fully developed flow and thermal fields within the vessel- the vessel- the vessel- the vessel-� neglecting end effects neglecting end effects neglecting end effects neglecting end effects ub—ave. blood flow velocity . blood flow velocity . blood flow velocity . blood flow velocity 0 , , , , , , , , , (0), (0) c k r r T u c k T T t t t b t b b b b b w ρ ρ ( ) ( ) T x T x w b Ave. (bulk) blood temp. Ave. (bulk) blood temp. Ave. (bulk) blood temp. Ave. (bulk) blood temp. Vessel wall temp. Vessel wall temp. Vessel wall temp. Vessel wall temp. q̇ = 0 Blood Flow Convection Blood Flow Convection Blood Flow Convection Blood Flow Convection Blood Flow Convection Blood Flow Convection Blood Flow Convection Blood Flow Convection

Blood Vessel segment Energy Balance: 9x =9com +9xtdr (1) If conduction in blood along the x-dir is neglected,then q,=im.CT(x)=p,·(uπ)CT(x) (2) 9a=P6·(4,π2)C6(x+dr(3) Rwm=万.2ibdk 万=ω店=2水。 T(x)- 2Tp To com Ro Reond (4) Rcond 2πk,·k

Energy Balance : Energy Balance : Energy Balance : Energy Balance : If conduction in blood along the x-dir is neglected, then If conduction in blood along the x-dir is neglected, then If conduction in blood along the x-dir is neglected, then If conduction in blood along the x-dir is neglected, then Blood Vessel segment Blood Vessel segment Blood Vessel segment Blood Vessel segment qx = qconv + qx+dx ( ) ( ) ( ) 2 q m C T x u r CT x x b b b b b b b b = ̇ = ρ ⋅ π ⋅ ( ) ( ) 2 q u r C T x dx x+dx = ρb ⋅ bπ b ⋅ b b + conv cond b conv R R T x T q + − = 0 ( ) k dx r r R r k r N k h h r dx R t b t cond b b b uD b b conv ⋅ = = ⋅ = ⋅ ⋅ = π π 2 ln( ) 2 2 2 1 (1) (2) (3) (4)

1 方=0=2 o 2rp To Where: In(分 2πk,·d Substituting Eq.(2),(3),(4)into (1), uwrip,CidTi(x) T(x)- =0 1,ln(/6) 4πk,·dk'2πk·dk 4() -((x)-) (T(x)-O) dr mn(/2) (5) u.P.C.k*2k 1+ L=uEP.C () 2k

Where: ) 2 ln( / ) 4 1 ( 2 t t b b b b b b b k r r k l = u r ρ C + 0 2 ln( / ) 4 1 ( ) ( ) 2 0 = ⋅ + ⋅ − + k dx r r k dx T x T u r C dT x t t b b b b b b b b π π π ρ k dx r r R r k r N k h h r dx R t b t cond b b b uD b b conv ⋅ = = ⋅ = ⋅ ⋅ = π π 2 ln( ) 2 2 , 2 1 Substituting Eq. (2),(3),(4) into (1), Substituting Eq. (2),(3),(4) into (1), Substituting Eq. (2),(3),(4) into (1), Substituting Eq. (2),(3),(4) into (1), b b t t b b b b b b b b l T x T k r r k u r C T x T dx dT x ( ( ) ) ) 2 ln( / ) 4 1 ( ( ) ( ( ) ) 0 2 0 − = − + − − = ρ (5)

From(5) T(x)-T=Ae (x=0)=T(0) T(x)-T=(T(0)-T)e (6) Referring to the circuit, 7(刊=79-9·R=70)- T()-1R R+R =T(x)-(T(x)-·B B= R.conv Where Rcom Rcond2 2k in( 1+- k

From (5) From (5) From (5) From (5) T x T x T B R R R T x T T x T x q R T x b b conv conv cond b w b conv conv b = − − ⋅ + − = − ⋅ = − ( ) ( ( ) ) ( ) ( ) ( ) ( ) 0 0 blx T b x T Ae − ( ) − 0 = Referring to the circuit, Referring to the circuit, Referring to the circuit, Referring to the circuit, (6) Where ( 0) (0) b Tb T x = = blx b b T x T T T e − ( ) − = ( (0) − ) 0 0 t b t b conv cond conv k r r k R R R B 2 ln( ) 1 1 + = + =

T(x)=T(x)1-B)+T·B ={[T(0)-e+(1-B)+BT T(x)=(1-B)[T(0)-Ie+T (T) Solution for Tr,T T3(x)=35.0+2.0e6.862x T(x=35.0+1.643e6.862r

0 0 0 0 {[ (0) ] }(1 ) ( ) ( )(1 ) T T e T B BT T x T x B T B blx b w b = − + − + = − + ⋅ − 0 0 T (x) (1 B)[T (0) T ]e T blx w = − b − + − (7) Solution for Solution for Solution for Solution for Tb(x), Tw(x) x w x b T x e T x e 6.862 6.862 ( ) 35.0 1.643 ( ) 35.0 2.0 − − = + = +

37.0- 36.5卡 T,(X)r。=500m T(x) ()'dweL 36.0 --T,N)r。-250 T(x) 35.5 35.0 0.0 0.2 0.4 0.6 0.8 1.0 X(m) Blood and vessel wall temperature distributions

Blood and vessel wall temperature distributions Blood and vessel wall temperature distributions Blood and vessel wall temperature distributions Blood and vessel wall temperature distributions 0.0 0.2 0.4 0.6 0.8 1.0 35.0 35.5 36.0 36.5 37.0 Temp.( oC) X(m) T b (x) rb=500 um T w (x) T b (x) rb=250 um T w (x)

Thermal Equilibrium Length A single vessel imbedded in a large tissue cylinder the length along which the temperature difference between the blood and tissue decays to (1/e)of the initial difference T(x=1)-=(T(0)-T)e≈36.79%(T(0)-) From previous analysis: T(x)-T=((0)-e 1=x+g2

the length along which the temperature the length along which the temperature the length along which the temperature the length along which the temperature difference between the blood and tissue difference between the blood and tissue difference between the blood and tissue difference between the blood and tissue decays to (1/e) of the initial difference decays to (1/e) of the initial difference decays to (1/e) of the initial difference decays to (1/e) of the initial difference Thermal Equilibrium Length Thermal Equilibrium Length Thermal Equilibrium Length Thermal Equilibrium Length Thermal Equilibrium Length Thermal Equilibrium Length Thermal Equilibrium Length Thermal Equilibrium Length A single vessel imbedded in a large tissue cylinder A single vessel imbedded in a large tissue cylinder A single vessel imbedded in a large tissue cylinder A single vessel imbedded in a large tissue cylinder A single vessel imbedded in a large tissue cylinder A single vessel imbedded in a large tissue cylinder A single vessel imbedded in a large tissue cylinder A single vessel imbedded in a large tissue cylinder e l ( ) ( (0) ) 36.79%( (0) ) 0 1 Tb x = l e − T0 = Tb − T0 e ≈ Tb − T − elx T x T T T e b b − ( ) − = ( (0) − ) 0 0 From previous analysis: From previous analysis: From previous analysis: From previous analysis: ) 2 ln( / ) 4 1 ( 2 t t b b e b b b b k r r k l = u r ρ C +

Thus: x=l f:PC6=4.16*10J1m.K,k,≈k,=0.5W1mK =10r% then: 1。=11.6588*10(4,6) An index for blood-tissue heat transfer

e x =l bCb = 4.16*10 J / m ⋅K, k b ≈ k t = 0.5W / m⋅K 6 3 ρ t b r =10r Thus: If: 11.6588*10 ( ) 6 2 e b b then: l = u r An index for blood-tissue heat transfer An index for blood-tissue heat transfer An index for blood-tissue heat transfer An index for blood-tissue heat transfer

VASCULAR SYSTEM DIMENSIONS Internal Length ( Systemic Total Thevmal Typical Diameter (mm) *Velocity Volume Volume E以bratio, Blood Vessel Number (range) (range (mn/9 (ml) Summary Lasth (lem) 1 10-30mm 300-650 500 156 146D00 Aorta 0.D2 460 Main branches 32 5-22mm 33-60 83 62o0 0.002 Large arteries 288 4-5 mm 14-28 134 104 Medium arteries 1152 2.5-4.0mm 10-22 117 Small arteries 3456 1.0-2.5mm 6-17 80 104 858 200 0.03 Tributaries 20,736 0.5-1.0mm 3-13 91 Small rami 82,944 250-500um 2-8 60 57 398 40 0,2 Terminal arteries 497,664 100-250um 1-6 25 5 4.6 1.3 Arterioles 18,579,456 25-100um 0.2-3.8 3.2 5 Metartenoles 238,878,720 10-25um 0.1-1.8 42 5500 Capillaries 1612443136 410画 0511 260 260 20f04 Postcapillary venules 4,408,161,734 8-30um 0.1-0.6 167 0.7 161 8*o3 556 Collecting venules 160,444,500 30-50um 0.1-0.8 1071 Muscular venules 32,088,900 50-100um 0.2-1.0 142 100-200m 0.5-3.2 13 330 15 2.1 Small collecting veins 10,241,508 Terminal branches 496,900 200-600um 1-6 207 3277 19,968 0.6-1.1mm 2-9 15 64 Small veins 16 o.1 512 1-5mm 10-20 67 Medium veins 256 5-9mm 1437 36 476 2206 2600 Large veins 0,905 Main branches 224 9-20mm 20-100 1538 1 20-35mm 200-500 3352 125 150o0 Vena cava 4395 heart 350 lungs 455 TOTAL 5200ml Schneck,DJ.An Outline of Cardiovascular Structure and Function,in The Biomedical Engineering Handbook. Bronzino,JD,Ed-in-Chief,CRC Press,Boca Raton,FL 1995,pp.3-14. Lightfoot,EN.The Roles of Mass Transfer in Tissue Function,in The Biomedical Engineering Handbook Bronzino,JD,Ed-in-Chief,CRC Press,Boca Raton,FL 1995,pp.1656-1670