【热力学】基于matlab模拟二维瞬态热传导允许用户分析随时间和空间变化的温度分布

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本文介绍了二维瞬态热传导模型在工程和科学研究中的重要性，详细讲解了其数学描述和数值求解方法，特别提到了使用Matlab提供的工具进行模拟，以可视化温度分布和分析不同材料的传热特性。

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🔥 内容介绍

热传导是一种重要的物理现象，它描述了热量如何在物质中传递。在许多工程和科学应用中，我们需要对热传导进行建模和分析，以便更好地理解和控制热力学过程。二维瞬态热传导是一种常见的热传导模型，它允许我们研究随时间和空间变化的温度分布。

二维瞬态热传导模型的基本假设是，物质内部的温度分布是二维的，并且随着时间的推移而变化。这个假设适用于许多实际应用，例如热处理、焊接和电子器件散热等。在这些应用中，我们需要了解物质内部的温度分布，以便更好地控制热力学过程，从而实现更好的性能和效率。

二维瞬态热传导模型的数学描述是一个偏微分方程，它描述了温度分布随时间和空间变化的规律。这个方程通常很难解析求解，因此需要使用数值方法进行求解。常用的数值方法包括有限元方法、有限差分方法和边界元方法等。这些方法可以有效地求解二维瞬态热传导模型，并给出物质内部温度分布的精确数值解。

二维瞬态热传导模型的应用非常广泛。例如，在热处理过程中，我们需要对材料内部的温度分布进行建模和分析，以便确定最佳的热处理参数。在焊接过程中，我们需要了解焊接区域的温度分布，以便控制焊接质量。在电子器件散热方面，我们需要对电子器件内部的温度分布进行建模和分析，以便设计更好的散热系统。

此 Matlab 提交提供了一个 2D 瞬态热传导仿真工具，用于分析不同长度和宽度的各种材料的传热。它使用户能够可视化随时间和空间的温度分布，并能够为指定位置创建温度与时间图表。

特征

- 2D 瞬态热传导模拟：模拟不同长度和宽度的不同材料的 2D 热传导

- 数值稳定性：确保稳定性检查并提供 ulerts

- 温度分布可视化：生成随时间和空间变化的温度分布图。

- 温度与时间图表：创建特定位置的温度随时间变化的图表。

- 定制：轻松配置材料属性和模拟参数。

- 用户友好的界面：直观的 GUI，可实现无缝交互。

总之，二维瞬态热传导是一个非常重要的物理现象，它允许我们研究随时间和空间变化的温度分布。通过数值方法求解二维瞬态热传导模型，我们可以得到物质内部温度分布的精确数值解。这些数值解对于许多工程和科学应用都非常有用，例如热处理、焊接和电子器件散热等。因此，二维瞬态热传导是一个非常重要的研究领域，值得我们进一步探索和研究。

📣 部分代码

function varargout = TwoDtransGUI(varargin)% TWODTRANSGUI MATLAB code for TwoDtransGUI.fig%      TWODTRANSGUI, by itself, creates a new TWODTRANSGUI or raises the existing%      singleton*.%%      H = TWODTRANSGUI returns the handle to a new TWODTRANSGUI or the handle to%      the existing singleton*.%%      TWODTRANSGUI('CALLBACK',hObject,eventData,handles,...) calls the local%      function named CALLBACK in TWODTRANSGUI.M with the given input arguments.%%      TWODTRANSGUI('Property','Value',...) creates a new TWODTRANSGUI or raises the%      existing singleton*.  Starting from the left, property value pairs are%      applied to the GUI before TwoDtransGUI_OpeningFcn gets called.  An%      unrecognized property name or invalid value makes property application%      stop.  All inputs are passed to TwoDtransGUI_OpeningFcn via varargin.%%      *See GUI Options on GUIDE's Tools menu.  Choose "GUI allows only one%      instance to run (singleton)".%% See also: GUIDE, GUIDATA, GUIHANDLES% Edit the above text to modify the response to help TwoDtransGUI% Last Modified by GUIDE v2.5 17-Apr-2016 02:20:13% Begin initialization code - DO NOT EDITgui_Singleton = 1;gui_State = struct('gui_Name',       mfilename, ...                   'gui_Singleton',  gui_Singleton, ...                   'gui_OpeningFcn', @TwoDtransGUI_OpeningFcn, ...                   'gui_OutputFcn',  @TwoDtransGUI_OutputFcn, ...                   'gui_LayoutFcn',  [] , ...                   'gui_Callback',   []);if nargin && ischar(varargin{1})    gui_State.gui_Callback = str2func(varargin{1});endif nargout    [varargout{1:nargout}] = gui_mainfcn(gui_State, varargin{:});else    gui_mainfcn(gui_State, varargin{:});end% End initialization code - DO NOT EDIT% --- Executes just before TwoDtransGUI is made visible.function TwoDtransGUI_OpeningFcn(hObject, eventdata, handles, varargin)% This function has no output args, see OutputFcn.% hObject    handle to figure% eventdata  reserved - to be defined in a future version of MATLAB% handles    structure with handles and user data (see GUIDATA)% varargin   command line arguments to TwoDtransGUI (see VARARGIN)% Choose default command line output for TwoDtransGUIhandles.output = hObject;% Update handles structureguidata(hObject, handles);% UIWAIT makes TwoDtransGUI wait for user response (see UIRESUME)% uiwait(handles.figure1);% --- Outputs from this function are returned to the command line.function varargout = TwoDtransGUI_OutputFcn(hObject, eventdata, handles) % varargout  cell array for returning output args (see VARARGOUT);% hObject    handle to figure% eventdata  reserved - to be defined in a future version of MATLAB% handles    structure with handles and user data (see GUIDATA)% Get default command line output from handles structurevarargout{1} = handles.output;function handles = getpara(handles)listStrings = get(handles.listbox1,'String');domaintype = listStrings{get(handles.listbox1,'Value')};switch domaintype    case 'Aluminium'        handles.data.alfa = 8.635e-5;    case 'Gold'        handles.data.alfa = 1.22e-4;    case 'Copper'        handles.data.alfa = 1.12e-4;    case 'Brass'        handles.data.alfa = 3.412e-5;    case 'Steel (0.5% C)'        handles.data.alfa = 1.474e-5;    case 'Steel (1.0% C)'        handles.data.alfa = 1.72e-5;    case 'Iron'        handles.data.alfa = 2.034e-5;    case 'Lead'        handles.data.alfa = 2.343e-5;    case 'Magnesium'        handles.data.alfa = 9.7e-5;endset(handles.mstxt,'String',...    sprintf('Material Property \nThermal Diffusivity of Selected Material \n\n%g (m2/sec)',handles.data.alfa));handles.data.plottime = str2double(get(handles.plottime,'String'));%GEOMETRY INPUTS%handles.data.lengthx = str2double(get(handles.lengthx,'String'));handles.data.lengthy = str2double(get(handles.lengthy,'String'));handles.data.chkx = str2double(get(handles.chkx,'String'));handles.data.chky = str2double(get(handles.chky,'String'));handles.data.nodex = str2double(get(handles.nodex,'String'));handles.data.nodey = handles.data.nodex;handles.data.dx = handles.data.lengthx/(handles.data.nodex-1);handles.data.dy = handles.data.lengthy/(handles.data.nodey-1);handles.data.chkndx = round(handles.data.chkx/handles.data.dx);handles.data.chkndy = round(handles.data.chky/handles.data.dy);set(handles.gitxt,'String',...    sprintf('Geometry \n DX= %g (m)\nDY= %g (m)\n\nMonitoring Position\n(x,y)= %g,%g',...    handles.data.dx,handles.data.dy,handles.data.chkndx,handles.data.chkndy));%TEMPERATURE INPUTS%handles.data.temp_initial = str2double(get(handles.tini,'String'));handles.data.temp_left_final = str2double(get(handles.tlf,'String'));handles.data.temp_right_final = str2double(get(handles.tlr,'String'));handles.data.temp_top_final = str2double(get(handles.tbt,'String'));handles.data.temp_bottom_final = str2double(get(handles.ttp,'String'));% TO CHECK MIN AND MAX TEMP FOR COLORBAR %handles.data.ctvec = [ handles.data.temp_initial handles.data.temp_left_final ...    handles.data.temp_right_final handles.data.temp_top_final handles.data.temp_bottom_final];handles.data.ctmax = max(handles.data.ctvec);handles.data.ctmin = min(handles.data.ctvec);% ---------------------------------------%TIME INPUTS in sec%handles.data.dt = str2double(get(handles.tstp,'String'));handles.data.total_time = str2double(get(handles.ttime,'String'));handles.data.convergance_criteria = str2double(get(handles.errv,'String'));handles.data.plotupdatetime = handles.data.dt*handles.data.plottime;set(handles.uptxt,'String',...    sprintf('Plot Update Time \n %g sec',handles.data.plotupdatetime));% INITIALIZATION %handles.data.fxx = (handles.data.alfa*handles.data.dt)/(handles.data.dx*handles.data.dx);handles.data.fyy = (handles.data.alfa*handles.data.dt)/(handles.data.dy*handles.data.dy);handles.data.fxy = handles.data.fxx+handles.data.fyy;set(handles.ttxt,'String',...    sprintf('Numerical Stability \nCurrent \nFourier Number = %g',handles.data.fxy));handles.data.ldx = 0:handles.data.dx:handles.data.lengthx;handles.data.ldy = 0:handles.data.dx:handles.data.lengthy;% NUMERICAL SSTABILITY CHECK %if handles.data.fxy > 0.5     warndlg({'Numerical Stability Condition: ',        sprintf('FOURIER NUMBER \n((DIFFUSIVITY*TIME STEP)*(1/(DX^2))+(1/(DX^2)))\n fo < 0.5'),        sprintf('Current Fourier Number = %g',handles.data.fxy)},'WARNING !!');end%MONITORING POSITION CHECK %if handles.data.chkndx > handles.data.nodex || handles.data.chkndy > handles.data.nodey    warndlg({'Monitoring Position Out of Bounds ',        sprintf('Place (X,Y) distance of monitoring position within the limits of specified dimensions\n 0 < x < %g\n 0 < y < %g'...        ,handles.data.lengthx,handles.data.lengthy)},'WARNING !!');endfunction twodconduction(handles)set(handles.stop,'UserData',0);alfa = handles.data.alfa;plottime = handles.data.plottime;lengthx = handles.data.lengthx;lengthy = handles.data.lengthy;chkx = handles.data.chkx;chky = handles.data.chky;nodex = handles.data.nodex;nodey = handles.data.nodey;dx = handles.data.dx;dy = handles.data.dy;chkndx = handles.data.chkndx;chkndy = handles.data.chkndy;temp_initial = handles.data.temp_initial;temp_left_final = handles.data.temp_left_final ;temp_right_final = handles.data.temp_right_final;temp_top_final = handles.data.temp_top_final;temp_bottom_final = handles.data.temp_bottom_final;ctvec = handles.data.ctvec ;ctmax = handles.data.ctmax ;ctmin = handles.data.ctmin;dt = handles.data.dt ;total_time = handles.data.total_time ;convergance_criteria = handles.data.convergance_criteria;plotupdatetime = handles.data.plotupdatetime ;fxx = handles.data.fxx;fyy = handles.data.fyy;fxy = handles.data.fxy;ldx = handles.data.ldx;ldy = handles.data.ldy;% INITIALIZING MATRIX %for i=1:nodex;    for j=1:nodey        t(i,j) = temp_initial;    endend % CALCULATION OF TEMPETATURE PROFILE %nt = 0;k = 1;shp = 0;x = 0:dx:lengthx;y = 0:dy:lengthy;[x,y] = meshgrid(x,y);while nt < total_time    if get(handles.stop,'UserData') == 1        break    end    if nt == 0        T = t;        T(1,:) = temp_top_final;        T(nodey,:) = temp_bottom_final;        T(:,1) = temp_left_final;        T(:,nodex) = temp_right_final;    end    for i = 2:nodey-1        for j = 2:nodex-1            T(i,j) = (t(i,j)*(1-(2*fxy)))+ ((fxx*(t(i-1,j)+t(i+1,j))+(fyy*(t(i,j-1)+t(i,j+1)))));            if i == chkndy                if j == chkndx                    Tchk(k) = T(i,j);                end            end        end    end    err = max(max((abs(T-t))));    tmax = max(max(T));    if err < convergance_criteria        % MESSAGE BOX %        msgbox({'STEADY STATE ACHIEVED',            sprintf('Time for steady state = %g',nt)},'RESULT');        k = k+1;        break    end    if shp == plottime        contourf(handles.contour,x,y,T,20);        caxis(handles.contour,[ctmin,ctmax]);        colorbar('peer',handles.contour);        xlabel(handles.contour,'DISTANCE x (m)'),        ylabel(handles.contour,'DISTANCE Y (m)'),        axis(handles.contour,'equal','tight'),        title(handles.contour,...            sprintf('TEMPERATURE PROFILE FOR TIME (SEC) = %g\nError = %g\nCheck Node Temp = %g',nt,err,Tchk(k)));        time = 1:k;        time = time.*dt;        plot(handles.axes2,time,Tchk)        grid on;        xlabel(handles.axes2,'TIME (sec)');        ylabel(handles.axes2,'TEMPERATURE (Celcius)');        title(handles.axes2,...            sprintf('TEMPERATURE PROFILE FOR NODE AT DISTANCE (m)\n = %g,%g',chkndx,chkndy));        pause(0.01)        shp = 0;    end    t = T;    nt = nt+ dt;    k = k+1;    shp = shp+1;endcontourf(handles.contour,x,y,T,20);caxis(handles.contour,[ctmin,ctmax]);colorbar('peer',handles.contour),xlabel(handles.contour,'DISTANCE x (m)'),ylabel(handles.contour,'DISTANCE Y (m)'),axis(handles.contour,'equal','tight'),title(handles.contour,...    sprintf('TEMPERATURE PROFILE FOR TIME (SEC) = %g\nError = %g\nCheck Node Temp = %g',nt,err,Tchk(k-1)));% --- Executes on button press in pushbutton1.function pushbutton1_Callback(hObject, eventdata, handles)% hObject    handle to pushbutton1 (see GCBO)% eventdata  reserved - to be defined in a future version of MATLAB