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 + = + =