热工基础珍藏版中文第9章导热

t=ft=f(x,y,z,ttt=f(x,y,z,ttt

t

t=f(x,y,t=f(x,y,t=f(x,

t ¶¶t=¶t„t=t=f(x(onedimensionalsteadystatetemperature

Dt,¶t,¶tD D Dx lim Dt:x Dxfi D

¶tngradt:

x

导热基本定律（Fourier’slawofheatFF=-lAgradt=-lA¶t¶q=-lgradt=-l¶t¶Φ热流量（heatflow）q热流密度（heatflux）¶t¶nn温度梯度（temperatureF=F=-lA¶t¶q=-l¶t¶

gradt=¶gradt=¶t¶xi+¶yj+¶z¶¶ q=-q=-lgradt=-l¶t¶=-l¶t-l¶ -l¶tjki¶¶=qxi+yj+qz

¶q=-l¶q=-l¶¶¶ =-l¶t

¶z =-l¶z¶t tq=F

tw1-t

W/m2 d

t = =

t = grad= gradW/m ，对大多数材料,l与t呈线形关系ll0（1bt ~、 、金属l2.2--420W/m.K影响：纯金属的温度t›,掺入杂质（合金）lfl（黄铜 l值：0.07—0.7假设：物体各向同性连续介质λ,ρ,с为常数，

фy ddFin+dFV-dFout=dU ф

dф

dфydф

dфdф

dфdFin+dFV-dFin+dFV-dFout=

yF

=-l¶ydxdz

Fz=-l

xxzX方向

x+dxy+dyz+dz

=-l¶(t+¶tdx)dydz =-l¶(t+¶tdy)dxdz =-l¶(t+¶tdz)dxdy¶z ¶z =FVdxdydzdU=rc¶tdxdydz

dFindFin+dFV-dFout=

dF+F+F+Fdxdydz- +F +F

rcrc¶t=[¶(l¶t)+¶(l¶t)+¶(l¶t)]+FVa=a== l(¶2t+¶2t+¶2t)+Faa= 数值范围：油1×10-7＿银2×104m2/s =a(¶2¶x+¶2¶y+¶z)FVrc==¶x+¶y+¶z2a(¶2¶xa(¶2¶x++¶y¶z)FVrc=¶2t+¶2t+¶2¶y¶z= rrj,zzx=rcosj y=rsinj, z=y rc¶t=1¶(lr¶t)+1¶(l¶t)+¶(l¶t)+F¶tr r ¶j d(rdt)=zθzθryxφ球坐标系中rqjx=rsinqcosj,y=rsinqsinj,z=rcosqrc¶t=

¶(l¶t)+

¶tr

r2sin2q¶j¶jr2sinq drdrd(r2dt)=drl¶2¶2¶2F¶trc=(++)+ ttw=f(x,y,z,ttw=consttw=const

tw(τ t(x,τt(x,τttt0Φδxtx=ttt0Φδxtx=d,tx=d=100t(t(x,τqw(τqqw=f(x,y,z,t qwqw=-l(¶t¶nw qqw=const绝热边界条件（Adiabaticqqw=(τt(x,τ(τt(x,τfh0x-l(-l(¶t =h( -twwfx=0,h1(tf1-

)=-l¶t(x,t)

ttf1tλttf1tλw1hhtw2Φtf20δxx=d,-

¶(x,¶(x,t)

=h2(tw2-tf2(τf(τft(x,τ0x t

0 t 0 t tt平壁的一tw1,tw2，且tw1>tw2

λλф

x λ0xλ0xф dxB.C x= t=x=dt=tw2t=C1x+C=t

1=-tt=ttdx00xф λdt=-tw1-tw2dx qq=-ldt=ltw1-tw2dxdQQ=-lAdt=lAtw1-tw2dxdqq=ltw1-d q=ltw1-tw2 =tw1-tw2

λфtt-tw1-tw2d

=ll

lK/

x ΦR

tqqDSRl=tw1-tw +d+

tw1A

λ λ λttw2tw3

tw4ф

δ

δ δ ΦttttqΦttttqn d

Rλ

w3Rλ w4tλtλ1λ2λ3 h1 δδ2δфxqqDtSR=t 1tf1-tfn+ d+t

tf2t

R

Rλ

RtRλ

Rλ

R

ttCλB1λt CλB1λt1B2λ t2B2λф t1Rλ

RλB1RλB2

t2Rλ

>10dtw1tw2，且tw1

tw1tt=f

tw2фr (rdt)= dr drB.C r= t=r= t=tw2t=C1lnr+C2

tw1 rdr

tw2фr =-tw1-tw2

ln2 2

2 2t=wt=w1-(tw1-w)ln(ln(r2rdrdt=drdt=-tw1-twln(r21r

rdr

фrqq=-ldt=ltw1-tw2drln(r21rq=q=ltw1-1ln(r2) tλtw1tw2фrrFF=Aq=-lAdtdr=(2prl) ltw1-twln(r21r=tw1-tw1tw1-tw1=ln(r2r1ln(d2d1

ф d tRl=1Rl=1ln(d2K/d1

2pllln(d21

ф R

d2 tFl=Fl=l=tw2plln(1d)1

ln( FF= tw1-tw in12pliln( i+1dd

tλw2λλw2λ3λλ1λrrr r4ttttttt

фrFl=i=1ntw1Fl=i=1ntw1-tw1ln( i+1did

Rλ

w3Rλ λλ λ2λфF= tf1-tf+ 1ln(di+1)1

FltFltf1-tf1n+1 1i+1)+ 0

r

tf2t

R

w3 w3λ λ λ

R

t

t

r

фr l=l0(1+bt d(ldt)= dx即是：d[l(1bt)dt

tw1

tw2 dx

tt+1bt2=-l02d- 2+t+t 2l=l=l0(1+bt)q=-ldt=-l(1+bt)dtdx dxq=l0 )[1+b d 或：q

tw1

-twd

l0(1+btmq

tw1-tw2d

=l0(1+btm =1 t λ λ λttA

ф δ

δ2δ

b>0b<0tb>0b<0tw2 λ2=0.11W/（m.K），δ2=5mm，中间石棉白云石瓦状 FDFDt=h1Atf1-tf ++1tf1 tf2Φ ,

ФФ

t∞λδ

Hx•ddx2

-F=ΦΦt∞hλФxδHlxB.C x= t=x=

dt=ФФ引入过余温度：θt

=t0-dx2ddx2d2t-F=•l肋片单位体积的散热量Φt∞hλ x δHlxΦt∞hλ x δHlx=UhqdxФФ•F Fc =Uhqdx=hUqAdx Adx F•将θFhUlAd2q=hUhUlAdx lA mΦt∞Φt∞hλФФx+dxδxdxHlx

ФФ x= ФФx= =dx

q=Cemx+Ce-mx dx=C1me

mx- me-mx2e-2

C1= emHC2= emH

+e-mHemH+e-mHemem(H-x)+e-m(H- +e-mHch[m(H-= ch(mHΦΦθ0ФxФHtθ0tθxθ∞H1 =q0ch(mH

•Negligibleheatlossfrom

θθ •Tiptemperatureθ0θ0θθ0FF=-lAdqdx=hlUAqth(mH0xh

hlUAq0th(mH)=th(mH0 hUHq mH0不锈钢实心圆杆的直径为10mm，长0.2mt0=120℃ фR Rc tw1-tw2 <

tw1-tw2

tΔt c c

R=-FBc R=-FBc„

HABCG瞬态导 = l(¶2t+¶2t+¶2t)+FtλΦtλΦrtλΦrλΦλtδδ t=t=ft=ft=ft∞t∞<t0,h。分析x‡0tf分析x‡0¶t=

tτ ττtτ ττ12τ3

£x<d,t>I. B. x=

t=t0¶t=

(0£X£d)x=d-l¶t=h(t-

- 引入过余温度：θttτtτ ττ12τ3 =

(0£x<d,t>0)B. x=

(0£X£d)x=d-l¶q=hq

- qq(x,t)=2sinqe-b2ncos(b x0bn+sinbcosnnn 特征值bn 方程tanb=Bi的根 n

qq(x,t)=t-t0-=f(Bi,Fo,xdtt=f(a,t,l,d,h,

Fo=at

t0- aFo增加时，q逐渐减小，t越接近于

Bi=hdl

Bi=hd1hd1h 1hBitτ Bitτ t01hττ12τtτ tτ0ττtττ123

tt=f

ttw=

tt=f诺模图的使用方法：

qq

q(x,t)q(x,t)=t-t0-=f(Bi,Fo,xdqq(x,y,t)=q(x,t)q(qq(x,r,t)=q(x,t)q(r,t)qq(x,y,z,t)=q(x,t)q(y,t)

Bi=hdl物理意义：Bi hdd

l=hd

tτ t τ 1τ1

BiBifi 数法（Lumpedparameter 物体内部温度分布：tfBi=l

求解t∞ t求解-rcVdt=hA(t-t -rcVdq=hAqdtI. =t0-