A Coupled PDE-ODE Model for Nonlinear Transient Heat Transfer with Convection Heating at the Boundary: Numerical Solution by Implicit Time Discretization and Sequential Decoupling
Abstract
:1. Introduction
2. Physical System and Mathematical Model
3. Dimensionless Equations and Formulation of the Mathematical Problem
4. Steady-State Analysis
5. Implicit Time Discretization and Sequential Decoupling
6. Solving the TPBVP via the FDM
6.1. Discretizing the TPBVP with the FDM
6.2. Solving the Nonlinear System via the Newton Method
6.3. Calculating the Elements of the Jacobian
7. Computer Experiments and Discussion
- Example 1
- Example 2
- Example 3
8. Conclusions
Author Contributions
Funding
Conflicts of Interest
Nomenclature
Symbol | Description | Unit |
$t$ | time | $\mathrm{s}$ |
$x$ | position | $\mathrm{m}$ |
$\left.u(x,t\right)$ | temperature in the solid body at position $x$ and time $t$ | $\mathrm{K}$ |
$T\left(t\right)$ | temperature in the tank and of the outgoing liquid at time $t$ | $\mathrm{K}$ |
${T}_{r}$ | temperature of the incoming liquid | $\mathrm{K}$ |
${T}_{0}$ | initial temperature in the tank and along the solid body | $\mathrm{K}$ |
$\kappa \left(u\right)$ | thermal conductivity of the solid body at temperature $u$ | $\mathrm{W}/(\mathrm{m}\xb7\mathrm{K})$ |
$\rho $ | density of the solid body | $\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$ |
${\rho}_{l}$ | density of the liquid | $\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$ |
${c}_{p}$ | specific heat capacity (at constant pressure) of the solid body | $\mathrm{J}/(\mathrm{k}\mathrm{g}\xb7\mathrm{K})$ |
${c}_{p,l}$ | specific heat capacity (at constant pressure) of the liquid | $\mathrm{J}/(\mathrm{k}\mathrm{g}\xb7\mathrm{K})$ |
$L$ | length of the solid body | $\mathrm{m}$ |
$A$ | area of the contact surface liquid—solid body | ${\mathrm{m}}^{2}$ |
$V$ | volume of the tank | ${\mathrm{m}}^{3}$ |
$Q$ | volumetric flow rate of the incoming/outgoing liquid | ${\mathrm{m}}^{3}/\mathrm{s}$ |
$H$ | mean convective heat transfer coefficient | $\mathrm{W}/({\mathrm{m}}^{2}\xb7\mathrm{K})$ |
$\mathsf{\Theta}\left(x,t\right)$ | $=\left(u\left(x,t\right)-{T}_{0}\right)/\left({T}_{r}-{T}_{0}\right)$, dimensionless temperature in the solid body | |
$\mathsf{\Psi}\left(t\right)$ | $=\left(T\left(t\right)-{T}_{0}\right)/\left({T}_{r}-{T}_{0}\right)$, dimensionless temperature in the tank | |
$\lambda \left(\mathsf{\Theta}\right)$ | $=\kappa \left(u\right)$, thermal conductivity of the solid body at $\mathsf{\Theta}$ | $\mathrm{W}/(\mathrm{m}\xb7\mathrm{K})$ |
${\lambda}_{0}$ | $=\lambda \left(0\right)$, thermal conductivity of the solid body at $\mathsf{\Theta}=0$ | $\mathrm{W}/(\mathrm{m}\xb7\mathrm{K})$ |
$g\left(\mathsf{\Theta}\right)$ | $=\lambda \left(\mathsf{\Theta}\right)/{\lambda}_{0}$, dimensionless thermal conductivity of the solid body at $\mathsf{\Theta}$ | |
${t}_{a}$ | $=V/Q$, characteristic time related to advection through the pipes | $\mathrm{s}$ |
${t}_{c}$ | $={\rho}_{l}{c}_{p,l}V/\left(HA\right)$, characteristic time related to convection in the tank | $\mathrm{s}$ |
$\overline{t}$ | $=\rho {c}_{p}{L}^{2}/{\lambda}_{0}$, characteristic time related to conduction in the solid body | $\mathrm{s}$ |
$\xi $ | $=x/L$, dimensionless position | |
$\tau $ | $=t/\overline{t}$, dimensionless time (Fourier number at $\theta =0$) | |
$\theta \left(\xi ,\tau \right)$ | $=\mathsf{\Theta}\left(x,t\right)$, dimensionless temperature in the solid body at $\xi $ and $\tau $ | |
$\psi \left(\tau \right)$ | $=\mathsf{\Psi}\left(t\right)$, dimensionless temperature in the tank at $\tau $ | |
$g\left(\theta \right)$ | $=g\left(\mathsf{\Theta}\right)$, dimensionless thermal conductivity of the solid body at $\theta $ | |
${\tau}_{a}$ | $={t}_{a}/\overline{t}$, dimensionless characteristic time related to advection | |
${\tau}_{c}$ | $={t}_{c}/\overline{t}$, dimensionless characteristic time related to convection | |
$\mathrm{B}\mathrm{i}$ | $=HL/{\lambda}_{0}$, Biot number at reference temperature $\theta =0$ |
Appendix A
function main mu=-2; N=51; dx=1/(N-1); M=51; tf=5; dt=tf/(M-1); ta=1; tc=1; Bi=1; R=(1/dt+1/ta+1/tc)^(-1); t=dt*(0:M-1)'; x=dx*(0:N-1)'; T=zeros(M,1); Y=zeros(N,M); F=zeros(N,1); J=zeros(N,N); J(N,N)=1; for m=2:M % time level y=Y(:,m-1); delta=1; while(delta>0.000001) % FDM + Newton g=exp(mu*y(1)); gy=mu*g; F(1)=g*(4*y(2)-3*y(1)-y(3))/2+dx*Bi*(R*(T(m-1)/dt+1/ta+y(1)/tc)-y(1)); F(N)=y(N); J(1,1)=gy*(4*y(2)-3*y(1)-y(3))/2-3*g/2+dx*Bi*(R/tc-1); J(1,2)=2*g; J(1,3)=-g/2; for i=2:N-1 g=exp(mu*y(i)); gy=mu*g; gyy=mu*gy; Dy=(y(i+1)-y(i-1))/(2*dx); f=((y(i)-Y(i,m-1))/dt-gy*Dy*Dy)/g; q=(1/dt-gyy*Dy*Dy-f*gy)/g; p=-2*gy*Dy/g; F(i)=y(i+1)-2*y(i)+y(i-1)-dx*dx*f; J(i,i-1)=1+dx*p/2; J(i,i)=-2-dx*dx*q; J(i,i+1)=1-dx*p/2; end dy=-J\F; y=y+dy; % Newton iteration delta=norm(dy,Inf); end T(m)=R*(T(m-1)/dt+1/ta+y(1)/tc); Y(:,m)=y; end plot3(0,0,1); hold on; xlim([-0.5 1]); ylim([0 tf]); zlim([0 1]); a=zeros(1,m); plot3([-.5+a;-.05+a],[t';t'],[T';T'],'k'); mesh(x,t,Y'); % plot3(-.5+a,t,T,'k'); % plot3(a,t,Y(1,:)','b'); % Flux_left=Bi*(T-Y(1,:)'); % plot3(a,t,Flux_left,'b'); % Flux_right=(-exp(mu*Y(N,:)).*(3*Y(N,:)-4*Y(N-1,:)+Y(N-2,:)))'/(2*dx); % plot3(1+a,t,Flux_right,'g'); end
Notation in Article | MATLAB Variable |
---|---|
$\mu $ | mu |
$M$ | M |
$N$ | N |
${\tau}_{f}$ | tf |
${\tau}_{a}$ | ta |
${\tau}_{c}$ | tc |
$\u2206\tau $ | dt |
$\u2206\xi $ | dx |
$\mathrm{B}\mathrm{i}$ | Bi |
$R$ | R |
$n+1$ | m |
$i$ | i |
${\tau}_{n}$ | t(m) |
${\xi}_{i}$ | x(i) |
${\psi}_{n}$ | T(m) |
${\theta}_{n,i}$ | Y(i,m) |
${\mathsf{\theta}}_{n}$ | Y(:,m) |
${\mathsf{\theta}}_{n-1}$ | Y(:,m-1) |
${\mathsf{\theta}}_{n}^{\left(k\right)}$ | y |
${\mathsf{\delta}\mathsf{\theta}}_{n}^{\left(k\right)}$ | dy |
${\mathbf{F}}_{n}\left({\mathsf{\theta}}_{n}^{\left(k\right)}\right)$ | F |
${\mathbf{J}}_{n}^{\left(k\right)}$ | J |
${\left|\right|{\mathsf{\delta}\mathsf{\theta}}_{n}^{\left(k\right)}\left|\right|}_{\infty}$ | delta |
${\theta}_{n,i}^{\left(k\right)}$ | y(i) |
$g\left({\theta}_{n,i}^{\left(k\right)}\right)$ | g |
${\partial}_{\theta}g\left({\theta}_{n,i}^{\left(k\right)}\right)$ | gy |
${\partial}_{\theta \theta}^{2}g$(${\theta}_{n,i}^{\left(k\right)}$) | gyy |
${\mathcal{D}\theta}_{n,i}^{\left(k\right)}$ | Dy |
$f({\theta}_{n,i}^{\left(k\right)},{\mathcal{D}\theta}_{n,i}^{\left(k\right)};{\theta}_{n-1,i})$ | f |
${q}_{n,i}^{\left(k\right)}$ | q |
${p}_{n,i}^{\left(k\right)}$ | p |
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$\mathit{l}$ | ${\mathit{N}}_{\mathit{l}}-1$ | ${\mathit{\psi}}_{\mathit{l}}$ | ${\mathit{E}}_{\mathit{l}}$ | ${\mathit{E}}_{\mathit{l}-1}/{\mathit{E}}_{\mathit{l}}$ |
---|---|---|---|---|
1 | 4 | 0.500861 | 0.001146 | |
2 | 8 | 0.500007 | 0.000292 | 3.93 |
3 | 16 | 0.499788 | 0.000074 | 3.97 |
4 | 32 | 0.499733 | 0.000018 | 3.99 |
5 | 64 | 0.499719 | 0.000005 | 4.00 |
6 | 128 | 0.499716 | 0.000001 | 4.04 |
$\mathit{l}$ | ${\mathit{N}}_{\mathit{l}}-1$ | ${\left.{\mathit{\theta}}_{\mathit{l}}\right|}_{\mathit{\xi}=0}$ | ${\mathit{e}}_{\mathit{l}}$ | ${\mathit{e}}_{\mathit{l}-1}/{\mathit{e}}_{\mathit{l}}$ |
---|---|---|---|---|
1 | 4 | 0.225643 | 0.001918 | |
2 | 8 | 0.224178 | 0.000452 | 4.24 |
3 | 16 | 0.223835 | 0.000109 | 4.15 |
4 | 32 | 0.223752 | 0.000027 | 4.08 |
5 | 64 | 0.223732 | 0.000007 | 4.05 |
6 | 128 | 0.223727 | 0.000002 | 4.07 |
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Filipov, S.M.; Hristov, J.; Avdzhieva, A.; Faragó, I. A Coupled PDE-ODE Model for Nonlinear Transient Heat Transfer with Convection Heating at the Boundary: Numerical Solution by Implicit Time Discretization and Sequential Decoupling. Axioms 2023, 12, 323. https://doi.org/10.3390/axioms12040323
Filipov SM, Hristov J, Avdzhieva A, Faragó I. A Coupled PDE-ODE Model for Nonlinear Transient Heat Transfer with Convection Heating at the Boundary: Numerical Solution by Implicit Time Discretization and Sequential Decoupling. Axioms. 2023; 12(4):323. https://doi.org/10.3390/axioms12040323
Chicago/Turabian StyleFilipov, Stefan M., Jordan Hristov, Ana Avdzhieva, and István Faragó. 2023. "A Coupled PDE-ODE Model for Nonlinear Transient Heat Transfer with Convection Heating at the Boundary: Numerical Solution by Implicit Time Discretization and Sequential Decoupling" Axioms 12, no. 4: 323. https://doi.org/10.3390/axioms12040323