ME 515 Midterm Exam Due Monday Nov. 2, 2009 (You have time, so please be neat and clear)

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1 ME 515 Midterm Exam Due Monday Nov. 2, 2009 (You have time, so please be neat and clear) A thermal burn occurs as a result of an elevation in tissue temperature above a threshold value for a finite period of time. The values of both the absolute temperature and the exposure time are crucial in determining the extent of injury. Temperature and time are not independent parameters affecting a burn; rather, clinical data indicate that a nonlinear coupling exists that fixes the severity of trauma. In general, the transient tissue temperature integrated over the time of exposure must be considered in creating a thermal lesion. The larger the value of this coupled integral, the greater the potential for injury. (For additional information, and the source of some of this information see: K. R. Diller, "Analysis of skin burns", in Heat Transfer in Medicine and Biology, Vol 2, Ed. Auraham Shitzer and R. C. Eberhart, Plenum Publishing Corp However, additional information is not required for this exam.) Numerous tests have been performed to determine the extent of burn injury from a temperature-time history. The damage function for burns has been expressed in terms of an Arrhenius relationship, as shown in Eq. 1, Table I, and Figure 1. T T 0 t t exp A 0 T T * (1)

2 Figure 1: Determination of Burn Injury from the Temperature-Time History Consider heat conduction in an x,y system which is governed by: T ( D T) S t where T is temperature expressed in C; the diffusivity, D, is K/c and has units of L 2 /time. The source term S has units of (Energy/unit area/unit time). The latter term is required since the most frequent cause of skin burns is associated with the application of a heat source. However, many times this heat source is applied as a boundary condition instead of an internal source

3 of heat. In that mode, the S term is zero and the boundary condition has a surface heat flux, q 0. A physical schematic of a skin layer is shown in figure 2 (Fig 3 from reference). The top layer is the epidermis which lies above the corium. At the bottom of the figure is the subcutaneous region. I have identified 4 thermally distinct regions in the figure with the necessary thermal properties listed in Table II. Figure 2: Depth of burn injury in skin: (I) Superficial burn; involves only the subcorneal layer. (IIA) Superficial partial thickness; involves some of basal layer. (IIB) Deep partial thickness; complete loss of the basal layer. (III) Full thickness; all epidermal elements are destroyed.

4 Table II: Thermal Properties of Skin Regions Region Material Diffusivity (cm 2 /s) 1 Corium (Epidermis also) 2 Subcutaneous Blood Vessels alpha * Sebaceous gland 0.05 Hair follicle The depth of this sample is 3.25 mm. One can assume no flux conditions on both sides of the specimen. The top and bottom surfaces will be subject to a series of boundary conditions. How to model the regions? This situation is a primary driving force for using finite elements! I have created a mesh that has Nx=35, Ny=30. Both directions include shadow node potentials. That is you do not necessarily need shadow nodes

5 I have also created a text file (grid.txt) that has 30 rows of data and 35 entries per row. The second row corresponds to y=0.0 mm (i.e. the surface). The second to last row is at a y = 3.25 mm depth. Assume delta_y = delta_x. Grid.txt

6 a.) Write a valid finite difference formulation for the governing equation using central finite difference expressions throughout. Convert your formulation to an ADI formulation, clearly showing the two step process for the implementation. b.) Develop the ADI numerical code to solve for the following boundary condition possibilities: 1.) Type I b.c. at top and bottom 2.) Type II b.c. at top Type I b.c. at bottom User-inputs specify the boundary condition options. REMEMBER, this is an ADI formulation of a parabolic PDE. You DO NOT iterate within a timestep. The w term added to your elliptic ADI formulation was a technique to converge for elliptic PDEs. The parabolic time derivative term acts as the w term previously. Since at t=0, all previous values are known, there are no iterations. One simply marches along in time. c.) For the following ICs and BCs determine when each of the first 3 burn levels occurs along the line labeled IIB. (See Table I and Fig. 1.) Line IIB is located at line 25 of the grid superimposed figure. T(x,y,0) = 37 C ; T(x,y=top,t>0) = 65 C ; T(x,y=bottom,t>0) = 37 C let alpha = 1 i.e. Run your program with dt = 25s. Report your thermal profiles at t = 0., 75., 250., and 500 seconds. (The Matlab contour function is helpful for this display.) Determine an approximate temperature along line IIB at t = 250s and use eq. 1 to estimate burn time. Once again, you will probably exceed a burn category before the skin section reached 250s. d.) Modify the value of alpha to 25, 50, 200 and repeat part c. Display the thermal field for alpha = 200 at t = 100s. Essentially, by increasing alpha one is giving an effective diffusivity to replace the advective component of the blood flow in the general advective-diffusive governing equation.

7 If one could integrate the transient temperature history at a node, one might be able to more accurately predict the extent of damage. Experimental data appears to support the follow damage integral: A A T exp( 2 ) 1 dt where A 1 and A 2 are discontinuous functions of temperature which provide an acceptable correlation with experimental data. A 1 = x s -1 ; A 2 = 5.0 x 10 4 K for 317 < T <= 323 (Kelvin) and A 1 = x s -1 ; A 2 = 8.0 x 10 4 K for 323 < T <= 333 (Kelvin) Using these values and expressions, First degree burns occur for > 0.1, Partial Second degree burns for > 1.0, Full Secondary burns for > 7500, and Third degree burns for > e.) Develop within your code a summation that measures for each node. Using the threshold damage values repeat (d) for alpha = 200 and compare your results. Report the time and temperature profile for damage values corresponding to first, partial second, and full second. In your code examine the nodes along (line 25). For the first two cases use dt = 0.1s. For the last case use dt = 250s. f.) Assume that there is a constant heat flux of 1 cal/(cm 2 s) applied to the surface of the skin. Repeat e with this boundary condition change. Use dt = 1.0s for the first two cases and dt = 50s for the last case. g.) Consider a heating pad that measures 1.5 ft x 1 ft and consumes 500 watts. If a patient lies on this pad is there a burn danger? Work out the situation and use your numerical skills to answer this problem.

8 Note: There are numerous ways to read a file into a Matlab program. This is only one strategy, but it might be helpful. % Read in data fid = fopen('grid.txt'); for j = ny:-1:1 for i = 1:nx G = fscanf(fid, '%d', 1); D(i,j) = diff(g+1); end end status = fclose(fid); You might have a diffusivity array D(i,j) but the input file has only 5 integers (0-4). So one could create an array of length 5 and store the 5 different diffusivity values, i.e. diff(g+1) locations. Then map those values for each node in the D(i,j) two dimensional array.

9 2.) Consider a 3-level time stepping scheme for a hyperbolic PDE U U t x 2 2 c Use central difference in time and central difference in space formulations. Recall, as in the 3-level parabolic stability analysis, that the spatial formulation is a function of. U U (1 ) U U 2 2 k k 1 k k 1 i i i i Perform a Fourier (Von Neumann) Stability analysis for this formulation. Be sure to express the stability constraints in terms of and c. What are the units of c? What non-dimensional number is constructed with c, delta_t, and delta_x? Be sure that your presentation is complete. Do not skip steps and show all work clearly.