Lecture 1: Newton s Cooling Law, Fourier s Law and Heat Conduction

Transcription

1 Lecture 1: Newton s Cooling Law, Fourier s Law and Heat Conduction V.Vuorinen Aalto University School of Engineering Heat and Mass Transfer Course, Autumn 2016 October 31 st 2017, Otaniemi ville.vuorinen@aalto.fi

2 Overview Part 1 ( must know ) Fourier's law Newton's law of cooling Application of Fourier's law to thermal resistance Heat equation in 1d Initial and boundary conditions Part 2 ( must know, mostly in the following lecture) Explicit Euler method, numerical time integration basics How to apply Newton s law of cooling in Matlab?

3 Part 1: Newton, Fourier and Conduction

4 Newton's cooling law Newton's cooling law: Rate of change of heat (W) for an object is proportional to temperature difference between the object and its surroundings. q=ha s (T s T ) Heat transfer coefficient T (t)=? T =const. [q]=j /s,[m]=kg,[c p ]=J /kg K,[T ]=K,[h]=W /m 2 K,[ A s ]=m 2 A s q= mc p dt dt =ha s (T T ) Note: most typically the cooling law takes T to be the object surface temperature. Lumped capacitance assumption (see Ch. 5) assumes T=uniform within the object.

5 Fourier's Law Heat flux vector q'' q' '= k T Note that units need to match! [q' ']=W /m 2,[T ]=K,[k]=W /m K,[ T ]=K /m thermal conductivity thermal diffusivity k=αρ c p density [c p ]=J /kg K,[ρ]=kg/m 3, [α]=m 2 / s specific heat [q]=w,q=q ' ' A Heat transfer rate

6 Fourier's Law Heat flux q'' in 1d q' ' = k dt dx = k Δ T Δ x Some benefits of Fourier s law: temperature drop across a layer temperature in a point between cold and hot temperature

7 Thermal Resistance for Conduction Consider a wall surrounded by hot and cold air Quite easily available: temperatures A and B, heat transfer coefficients, and thermal conductivities We want to know many things, e.g. 1) How many J/s is going through the wall? 2) What is the temperature in the middle of the wall? Convective heat transfer coefficient T A, h A T (x) L T B,h B T A, h A T 1 T 2

8 q 1 Derivation of Heat Flux Through a Single-Layered Wall Steady state: q 1 =q 2 =q 3 =q q 3 q /(h A A)=(T A T 1 ) (1) [q]=w, q=q ' ' A q/(k/ L)=(T 1 T 2 ) (2) q 2 A=area of cross-section q/(h B A)=(T 2 T B ) (3) Sum: (1)+(3) and note that T 1 -T 2 appears i.e. (2) T A T B +T 2 T 1 =q /(h A A )+q /(h B A) q= T A T B 1/(kA /L)+1/(h A A)+1/(h B A) Note: Generalizes to multilayered materials with various thicknesses!

9 For Multiple Material Layers Overall heat transfer coefficient q= A Δ T Σ i 1/(k i / L i )+1/(h A )+1/(h B ) U = 1 R tot A Thermal resistance Some benefits of thermal resistance concept: design of thermal insulation (buildings, clothes, combustion) allows to maximize or minimize heat flux allows designs to avoid hot pools of temperature from forming allows to design temperature profiles (e.g. avoid condensation)

10 Heat Equation without Source Terms 1d ρ c p T t = x (k T x ) T t = x (α T x ) In general T t = α T Heat equation in 2d with all terms written T t = x k T x + y k T y Task is to solve for the unknown temperature T = T(x,y,z,t). Initial conditions needed Boundary conditions needed But how can we solve partial differential equations sometimes analytically numerically

11 What does Heat Equation Mean? Task is to solve for the unknown temperature T = T(x,y,z,t). Fourier s law: W/m 2 ρ c p T t = x T (k x )= q q out in Δ x Derivation: 1/m Rate of change of internal energy per volume: W/m 3 Heat conduction and Fourier s law determine at which rate energy changes in a given point and time. Think: how many Joule s leave a small volume, how many arrive in the volume within time dt.

12 One of the Key ILO s on the Course is the Heat Sink Problem: Heat Sinks Often Used to Transfer Heat Away from Electronics Fin 1 Fin 2 y z Fin 19 x Heat equation inside the solid material (aluminium). What boundary conditions? Navier-Stokes equations in the gas, primary direction of convection is z. What boundary conditions?

13 Part 2: Basic numerical methods in heat transfer

14 Overview Numerical methods have been mathematically developed in order to solve analytical, mathematical equations approximately by computer. Examples: Taylor series, Newton's method, Solution of Ax = b, Iteration schemes The problem: analytical solution by pen and paper of various equation types can be very difficult and has of severe limitations. It is vary convenient to understand the very basics of numerical solution of heat equations in 0d, 1d and 2d. Numerical software like Matlab and Python could be used for short implementation of numerical solvers which give e.g. temperature as a function of space and time. In particular, for complex geometries and combined heat/fluid flow, various CFD (computational fluid dynamics) software like OpenFOAM, ELMER, Fluent etc can be used. Using the Matlab programs (supplementary course material) you learn that basics of heat transfer can be simulated with very short code of only a couple of lines. Learn basic ideas behind numerical solution on the course for future work life! extend your opportunities

15 Newton's cooling law Newton's cooling law: Rate of change of heat (W) for an object is proportional to temperature difference between the object and its surroundings. Usually the heat transfer coef. h unknown and object irregular shape q= mc p dt dt =ha s (T T ) A s T (t)=? T =const. [q]=j /s,[m]=kg,[c p ]=J /kg K,[T ]=K,[h]=W /m 2 K,[ A s ]=m 2

16 Newton's cooling law Rearrange the terms dt dt = ha s c p m (T T ) Initial condition T (t=0)=t o Analytical solution by separation of variables T (t)=(t o T )exp( ha s c p m t)+t Analytical solution exists good starting point for the computer learning: how to numerically solve temperature development in the above equation?

17 Solving temperature numerically over one timestep dt dt = ha s c p m (T T ) dt = dt ha s c p m (T T ) Note: explicit Euler method here (accuracy 1 st order in time; O(dt) ). Alternatives could be e.g.: 2 nd, 3 rd, 4 th order Runge-Kutta methods Δ T = Δ t ha s c p m (T T ) Δ T n = Δ t ha s c p m (T n T ) T n+1 =T n +Δ T n Solution proceeds in discrete timesteps t n =nδ t,n=0,1,2,...

18 This transforms into the following pseudo-code Step 0:T o known Step 1: Δ T n = Δ t ha s c p m (T n T ) Step 2:T n+1 =T n +Δ T n Step 3: goto Step1 until simulation time exceeded

19 Time-evolution of temperature in a refridgerator (link to homework 1a/5)

20 This Scheme is Extremely Short to Program in Matlab Program: /Example0d/cool0d.m Execution: >> cool0d What it does: Solves 0d Newton's cooling law for temperature of a 0d drink can. Snapshot of code that does the job: cp = 4190; % specific heat J/kgK dt = 20; % timestep in s To = ; % initial temperature K Tinf=273+4; % fridge temperature K simutime = 3*3600; % simulation time s simusteps = round(simutime/dt); T = To; % initial temperature for(k=1:simusteps) dt = -(h*as/(m*cp))*dt*(t-tinf); T = T+dT; Tcol(k) = T; % collect temperatures to Tcol end HOW IN PRACTICE TO IMPLEMENT THIS? open Matlab terminal open text editor create new file with name cool0d.m add the text from the left to file cool0d.m run by typing text cool0d on terminal

21 cp = 4190; % specific heat J/kgK dt = 20; % timestep in s To = ; % initial temperature K Tinf=273+4; % fridge temperature K simutime = 3*3600; % simulation time s simusteps = round(simutime/dt); T = To; % initial temperature for(k=1:simusteps) dt = -(h*as/(m*cp))*dt*(t-tinf); T = T+dT; Tcol(k) = T; % collect temperatures to Tcol end Example for N+1 = 5

22 Plotting the results 1) e.g. plot(x,y,'k-') where x and y are vectors 2) Note: length(x)=length(y) figure(1), clf, box, hold on 3) >> help plot alltime = linspace(0,simutime/3600, simusteps); plot(alltime, Tcol, 'k-','linewidth',2) plot(alltime, (To-Tinf)*(exp(-h*As*3600*alltime/(m*cp))) + Tinf, 'b--','linewidth',2) plot(alltime, (273+7)*(ones(length(alltime),1)), 'r-', 'Linewidth',2) plot(alltime, (Tinf)*(ones(length(alltime),1)), 'g-', 'Linewidth',2) h=xlabel('time (h)'); h=ylabel('temperature (K)'); h=legend('numerical solution', 'Analytical solution','t = 7 deg C','T = 4 deg C'); set(h,'fontsize', 16) print -dpng TcoolingCan Example for N+1 = 5

23 The Following Folders and Files for Matlab Programs Provided (Week 1) /Example0d/ cool0d.m /Examples1d/ HeatDiffusion.m ConvectionDiffusion.m /HowToPlot/ DrawingSurface.m PlottingFigure.m SurfaceAnimation.m /Examples2d/ CaseDefinition.m computedt.m GradX.m GradY.m HeatDiffusion2d.m project.m solvetemperature.m circle.m DivDiv.m GradXskew.m GradYskew.m Laplacian.m settbcs.m visualizeresults.m Execute by >> cool0d Execute by e.g. >> HeatDiffusion Demos on plotting Figures. Execute by e.g.: >> DrawingSurface 2d heat transfer code. Execute by: >> HeatDiffusion2d

24 Thank you for your attention!