微元满足能量守恒定律， [ t , t + d t ] [t, t+dt] [t,t+dt]

外界流入热量 + 内部热源产热 = 温度升高所需热量 Q 流 入 + Q 热 源 = Q 温 度 升 高 Q_{流入}+Q_{热源}=Q_{温度升高} Q流入​+Q热源​=Q温度升高​ 傅里叶热传导定律：热量从高温处向低温处流动，沿某方向流动热量的多少与温度在该方向的减少率成比例。 q → = − k ∇ u = { q x = − k ∂ u ∂ x q y = − k ∂ u ∂ y q z = − k ∂ u ∂ z \overrightarrow q=-k\nabla u = \begin{cases} q_x = -k\frac{\partial u}{\partial x} \\ q_y = -k\frac{\partial u}{\partial y} \\ q_z = -k\frac{\partial u}{\partial z} \end{cases} q ​=−k∇u=⎩⎪⎨⎪⎧​qx​=−k∂x∂u​qy​=−k∂y∂u​qz​=−k∂z∂u​​ 其中 q → \overrightarrow q q ​是热流密度矢量，表示单位时间沿单位面积的法向流出的热量。

∴ \therefore ∴ Q 左 右 = q ∣ x ⋅ d t ⋅ d y d z − q ∣ x + d x ⋅ d t ⋅ d y d z = ( q ∣ x − q ∣ x + d x ) ⋅ d t d y d z = − ∂ q ∂ x d x ⋅ d t d y d z = − ∂ ∂ x ( − k ∂ u ∂ x ) ⋅ d t d V = k ∂ 2 u ∂ x 2 d t d V Q_{左右}=q|_x·dt·dydz-q|_{x+dx}·dt·dydz \\ =(q|_x-q|_{x+dx})·dtdydz \\ =-\frac{\partial q}{\partial x}dx·dtdydz =-\frac{\partial}{\partial x}(-k\frac{\partial u}{\partial x})·dtdV \\ =k\frac{\partial^2u}{\partial x^2}dtdV Q左右​=q∣x​⋅dt⋅dydz−q∣x+dx​⋅dt⋅dydz=(q∣x​−q∣x+dx​)⋅dtdydz=−∂x∂q​dx⋅dtdydz=−∂x∂​(−k∂x∂u​)⋅dtdV=k∂x2∂2u​dtdV

Q 前 后 = q ∣ y ⋅ d t ⋅ d x d z − q ∣ y + d y ⋅ d t ⋅ d x d z = ( q ∣ y − q ∣ y + d y ) ⋅ d t d x d z = − ∂ q ∂ y d y ⋅ d t d x d z = − ∂ ∂ y ( − k ∂ u ∂ y ) ⋅ d t d V = k ∂ 2 u ∂ y 2 ⋅ d t d V Q_{前后}=q|_y·dt·dxdz-q|_{y+dy}·dt·dxdz \\ =(q|_y-q|_{y+dy})·dtdxdz \\ =-\frac{\partial q}{\partial y}dy·dtdxdz = -\frac{\partial}{\partial y}(-k\frac{\partial u}{\partial y})·dtdV \\ =k\frac{\partial^2 u}{\partial y^2}·dtdV Q前后​=q∣y​⋅dt⋅dxdz−q∣y+dy​⋅dt⋅dxdz=(q∣y​−q∣y+dy​)⋅dtdxdz=−∂y∂q​dy⋅dtdxdz=−∂y∂​(−k∂y∂u​)⋅dtdV=k∂y2∂2u​⋅dtdV

Q 上 下 = k ∂ 2 u ∂ z 2 ⋅ d t d V Q_{上下}=k\frac{\partial^2 u }{\partial z^2}·dtdV Q上下​=k∂z2∂2u​⋅dtdV

Q 流 入 = k ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 ) ⋅ d t d V = k Δ u ⋅ d t d V Q_{流入}=k(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2})·dtdV=k\Delta u·dtdV Q流入​=k(∂x2∂2u​+∂y2∂2u​+∂z2∂2u​)⋅dtdV=kΔu⋅dtdV

∵ \because ∵ Q 热 源 = g ( x , y , z , t ) d t d V Q_{热源}=g(x,y,z,t)dtdV Q热源​=g(x,y,z,t)dtdV g ( x , y , z , t ) g(x,y,z,t) g(x,y,z,t)表示单位体积内部热源的产热率（单位时间单位面积产热量） Q 升 温 = c ⋅ ρ d V ⋅ [ u ( t + d t , x , y , z ) − u ( t , x , y , z ) ] = c ⋅ ρ d V ⋅ ∂ u ∂ t d t = c ρ ∂ u ∂ t d t d V Q_{升温}=c·\rho dV·[u(t+dt,x,y,z)-u(t,x,y,z)] \\=c·\rho dV·\frac{\partial u}{\partial t}dt \\=c\rho \frac{\partial u}{\partial t}dtdV Q升温​=c⋅ρdV⋅[u(t+dt,x,y,z)−u(t,x,y,z)]=c⋅ρdV⋅∂t∂u​dt=cρ∂t∂u​dtdV 由能量守恒定律得 k Δ u ⋅ d t d V + g ( x , y , z , t ) d t d V = c ρ ∂ u ∂ t d t d V k c ρ Δ u + g ( x , y , z , t ) c ρ = ∂ u ∂ t k\Delta u·dtdV + g(x,y,z,t)dtdV = c\rho \frac{\partial u}{\partial t}dtdV \\\frac{k}{c\rho}\Delta u+\frac{g(x,y,z,t)}{c\rho}=\frac{\partial u}{\partial t} kΔu⋅dtdV+g(x,y,z,t)dtdV=cρ∂t∂u​dtdVcρk​Δu+cρg(x,y,z,t)​=∂t∂u​

热传导方程（扩散方程）: ∂ u ∂ t = a 2 Δ u + f ( t , x → ) ,   a = κ c ρ , f ( t , x → ) = g ( t , x → ) c ρ \frac{\partial u}{\partial t}=a^2\Delta u+f(t,\overrightarrow x),\space a=\sqrt{\frac{\kappa}{c\rho}},f(t,\overrightarrow x)=\frac{g(t,\overrightarrow x)}{c\rho} ∂t∂u​=a2Δu+f(t,x ), a=cρκ​ ​,f(t,x )=cρg(t,x )​ 其中， κ \kappa κ为热扩散系数。

量纲分析： [ k ] [ c ] ⋅ [ ρ ] = J / ( s ⋅ m ⋅ K ) J / ( k g ⋅ K ) ⋅ k g / m 3 = m / s [ u ] [ t ] = [ a 2 ] ⋅ [ u ] [ x 2 ]    ⟹    [ a 2 ] = m 2 / s \sqrt{\frac{[k]}{[c]·[\rho]}} = \sqrt{\frac{J/(s·m·K)}{J/(kg·K)·kg/m^3}}=m/\sqrt{s} \\ \frac{[u]}{[t]}=[a^2]·\frac{[u]}{[x^2]} \implies [a^2]=m^2/s [c]⋅[ρ][k]​ ​=J/(kg⋅K)⋅kg/m3J/(s⋅m⋅K)​ ​=m/s ​[t][u]​=[a2]⋅[x2][u]​⟹[a2]=m2/s 根据量纲可知，扩散传播距离与时间之间的关系： x 2 ∝ a 2 t x^2 \propto a^2t x2∝a2t