ME 320L/354L Laboratory 3

Transcription

1 ME 320L/354L Laboratory 3 Verification of numerical solution for transient conduction Introduction As we did last time around, we put on Fourier s hat, and his shoes for good measure, and pretend that we are verifying a new theory that we developed for transient conduction. To this end, we again use grandpa Hilton s apparatus (the same that was used for Laboratory Experiment 1), together with some other devices for automated data acquisition (two LeCroy digital scopes and the Hilton data acquisition module). The idea is that we can use a detailed temperature history collected at two boundary surfaces (T 1 and T 8 ), and known initial conditions, to calculate the temperature at any point in between, at any time. The calculation is essentially the solution of the heat equation by a finite difference technique. to verify the model, we also collect temperatures (T 2, T 3, T 6, T 7 ) at interior locations at various times, so that we can compare these with the results of our calculations. T1 T2 T3 T6 T7 T8 T T t t Theory The new model that we are proposing is that, in the case of a 1-D problem (such as temperature variation in a wall which is thin compared to its dimensions, or a rod insulated everywhere except for its ends), the temperature T (x, t) evolves with time according to the differential equation: dt dx = T α 2 x 2, (1) where α = k/(ρc). Since we are accomplished mathematicians, we know that in order to solve this problem we need an initial condition and a set of two boundary conditions, one for each end. These are, respectively, T (x, 0) = T i (x), (2) T (0, t) = T 1 (t), (3) T (L, t) = T 8 (t). (4) Before going ahead with our solution, we make our life easier (at least we like to think so) by nondimensionalizing the position, time and temperature as: ˆx = x L, (5) ˆt = αt L 2, (6) θ = T T A T B T A, (7) where L is some characteristic dimension of our wall or rod and T A and T B are some convenient reference temperatures (we will see later what works for this particular case). By doing this, we can re write the heat equation as: dθ dˆt = 2 θ ˆx 2. (8)

2 The boundary conditions are simply θ(0, ˆt) = θ 1 (ˆt), (9) θ(1, ˆt) = θ 8 (ˆt). (10) We propose to solve this using a finite difference (FD) algorithm. We represent the continuous temperature profile in the domain ˆx [0, 1] at time t p by a discrete set of temperatures (θ p 0, θp 1,..., θp n) evaluated at (n + 1) evenly spaced locations (x 0, x 1,..., x n ), as shown below: p p θ i-1 θ i p θ i+1 θ i p+1 x i-1 x i x i+1 At each location x i, we would like to find the value of the temperature at the next timestep t p+1, namely. Using a first order Taylor expansion, we can estimate this by: θ p+1 i θ p+1 i θ p i + dθ dˆt ˆt 2 θ ˆx 2 ˆt. (11) x=xi The second derivative of temperature with respect to position can be obtained using the following: 2 θ ˆx 2 θ i 1 2θ i + θ i+1 x=xi ˆx 2 (12) In summary, the FD algorithm consists of the following steps: 1. Obtain values of 2 θ/ ˆx 2 at each node location x i, using Eq. 12; 2. Update temperature profile using Eq. 11; 3. Increment current time ˆt; 4. If ˆt < ˆt max go to step 1 for new iteration; 5. Output results and stop program. The listing of a FORTRAN90 code that performs the above steps is given below: program fdexp simple 1-D finite difference code to treat conduction with specified temperature conditions at ends - explicit version implicit none real,dimension(:),allocatable :: x,theta,d2th_dx2,tout, > t1,t2,th1,th2 real bi,fo,dt,dx2,now

3 real left,right integer n,nsteps,i,j,k,npts character*24 t1fl,t2fl write(6,*) enter Fo read(5,*) fo note: logical units 5 and 6 are the standard input and output devices (keyboard and monitor) in FORTRAN write(6,*) enter number of steps read(5,*) nsteps write(6,*) enter temperature history file for T1 read(5,*) t1fl write(6,*) enter temperature history file for T8 read(5,*) t2fl open(11,file=t1fl,form= formatted ) open(12,file=t2fl,form= formatted ) read(11,*) npts read(12,*) npts allocate(t1(npts)) allocate(t2(npts)) allocate(th1(npts)) allocate(th2(npts)) do i=1,npts read(11,*) t1(i),th1(i) read(12,*) t2(i),th2(i) close(11) close(12) allocate(x(60)) subdivide domain into 59 pieces ( = 60 nodes ) do i=1,60 x(i)=(i-1)/60. allocate(theta(60)) same number of temperature values theta(1:30)=1.0 set initial conditions for left half-bar theta(1:30)=0.0 set initial conditions for right half-bar dt=fo/nsteps set times for output and open output file open(unit=11,form= formatted,file= fd.txt ) allocate(tout(51)) write(6,*) output at: do k=1,51 tout(k)=fo*exp((2*k-102)/10.) tout(k)=fo/50*(k-1) write(6,*) k,tout(k) dx2=(1/60.)**2 allocate(d2th_dx2(60)) k=1 write(6,*) beginning time loop... do n=1,nsteps now=(n-1)*dt left temperature

4 theta(1)=left(now,t1,th1) right temperature theta(60)=right(now,t2,th2) treat interior nodes do i=2,59 d2th_dx2(i)=(theta(i+1)-2*theta(i)+theta(i-1))/dx2 update temperatures theta(2:59)=theta(2:59)+d2th_dx2(2:59)*dt if (abs(now-tout(k)) <= dt) then write(6,*) output at,n*dt do j=1,60 write(11,10) n*dt,x(j),theta(j) write(11,*) k=k+1 end if close(11) output results write(6,*) temperature profile at Fo=,nsteps*dt do i=1,60 write(6,*) x(i),theta(i) 10 format(3e13.5e2) end real function left(now,t,th) this function generates a temperature history for the left side implicit none integer i real now real,dimension(1) :: t,th i=1 do while(.not.((now >= t(i)).and.(now < t(i+1)))) i=i+1 left=(th(i+1)-th(i))*(now-t(i))/(t(i+1)-t(i))+th(i) return end real function right(now,t,th) this function generates a temperature history for the right side implicit none

5 integer i real now real,dimension(1) :: t,th i=1 do while(.not.((now >= t(i)).and.(now < t(i+1)))) i=i+1 right=(th(i+1)-th(i))*(now-t(i))/(t(i+1)-t(i))+th(i) return end This code can be paraphrased in your favorite programming language, for example C, C++, Pascal, BASIC, or even Matlab or Mathematica. Lab procedure You will be collecting transient data, so it is very important to know exactly what to do before starting the experiment, otherwise you may need to spend much time waiting for the apparatus to return to the desired initial conditions before you can restart the experiment. If all goes well, the entire acquisition process should take 90 minutes or so (or F o = 17.5 if you are a brass rod m long). The following steps are suggested: 1. Set up the linear heat transfer apparatus by ensuring that thermocouples 1, 2, 3, 6, 7, 8 are connected to the control unit and to the oscilloscopes (1, 2, 3 on scope A, 6, 7, 8 on scope B). Set the water temperature on the recirculating bath to 5 C above zero. Also ensure that the mating ends of the brass rod are coated with an adequate amount of conductive paste. 2. Set up the control unit: With the two halves of the bar separated, adjust the voltage of the heater to 12V. This will begin heating up the half bar corresponding to temperatures 1, 2, and 3. When the temperature has reached about 100 C (this will take some time) turn the voltage down to 8V. 3. The data acquisition module will be used for gathering long-term transient temperature profiles. Its operation is quite simple: ensure that the computer is turned on. Then, turn on the acquisition module (the stainless steel clad box, HC110A) and the control module, H110. From the Programs menu on the PC, start the HC110 program. A screen will appear which allows you to choose the experiment you want to conduct:

6 Select HT110A - linear heat conduction ; you will then be asked to select from a range of experiments possible with the linear heat conduction unit. For the purpose of this lab, the most useful experiment is option 1, Temperature distribution through uniform plant wall ; You will be asked to enter a name for the data file, with a choice of four characters. You might want to use the date, say 1012, as the file name. Characters will be attached to the four you choose, so that

7 the full file name will become A.2. You will want to record the transients as they happen, so choose the option record data at set interval in the next menu; click on the go button, and you will see a screen that describes the physical layout of the experiment. Press tha Start button, and data acquisition will begin. After a short interval, you should see temperature data appear next to the experiment diagram: By clicking on the Graph and Table tabs, you will obtain screens with a plot of the temperature profile along the brass rod, and in tabular form, respectively. You will notice that three of the temperatures are uniformly hot, while the others are uniformly cool.

8 The Hilton acquisition unit is a bit slow to record data during the very initial stages of the process. For this reason we use the scopes to obtain transient data for the initial few minutes, where much of the

9 interesting stuff happens. The thermocouples for the hot half of the brass rod are plugged into scope A, while those for the cool half are plugged into B. Trace 4 in the scopes is used to synchronize the two traces. Now for the action bit: read this before beginning the experiment. Press the Start button on both scopes in case acquisition is not already under way. Wait for acquisition to slowly start. Put in a synchronizing signal on trace 4 by extracting and inserting the banana plug labeled as sync. At the same time, someone should read the time on the Hilton acquisition screen and note it down. Now take the two half rods and put them together with deliberate action, then secure them using the clips. Monitor the scope signal, and stop acquisition when the synchronizing trace is about to exit the screen. Continue to acquire data with the Hilton module until steady state is reached. Finally, after ending the data acquisition and quitting the HC110 program, copy the data files from the scopes and the computer to some floppy disks for later analysis. Analysis The test section is made of brass. Its thermal properties are: k = 121 Wm 1 K 1, ρ = 8450 kg m 3, c = 400 J kg 1 K 1. First, you must put your time histories for T1 and T8 in a spreadsheet. Then proceed to nondimensionalize the time and temperature signal using Eqns. 6 and 7. Non-dimensionalize the temperature using the initial temperature at T8 and the initial temperature at T1. Save the results as tab delimited text with the number of data points as the header. Also make sure that the time signals from the scopes are synchronized (you may need to shift the time by some amount from one of the two traces). Using variations of the code provided (note: this can be downloaded from: stuff/lab/lab.html), or a similar finite difference code if you like, calculate the temperatures at the same times and locations where measurements were taken, and compare them with the measurements. For each time data were taken, plot model results vs. measured results. Comment on accuracy (or otherwise) of the calculation, including possible reasons for deviation