3 - Elementar Solutions to Partial Differential Equations Most students believe that the term elementar solutions to partial differential equations is either an omoron or an irritating erudite epression better restated when dealing with college students. It seems there should be nothing elementar about solving a partial differential equation. herefore, it bears repeating that for a given partial differential equation preferabl one the student has derived and understands it is a fairl simple algebraic matter to generate a solution to the equation using a worksheet. Of course, the solution gets more difficult as the number of independent variables increases. For eample, the two directions on a worksheet (up-down and right-left) could be used to represent two independent variables. Figure 1 is a guide showing schematicall how a worksheet ma be construed for selected partial differential solutions. picall, in one-dimensional unstead state heat conduction problems (1D USS HC), time is made to increase down the sheet while the position through the solid is represented across the sheet. his forms a table in which the computed temperatures at an time and position ma be reported. In a twodimensional stead state heat conduction problem (D SS HC), the two directions on the sheet are used to represent the two dimensions in the conduction problem. A two dimensional unstead state heat conduction problem (D USS HC) has three independent variables (t,, ). here is no wa to represent all of these on the two-dimensional field offered b the worksheet; therefore, some accommodation must be made. picall this accommodation consists of using the two dimensions on the worksheet for the two independent dimensions ( and ). Each panel of temperatures on this area represents a particular time during the solution starting at the initial time and progressing to the final time as desired b the user. A three dimensional problem must be displaed as numerous slices through the solid be the solution stead state or unstead state. 3D USS HC (t,,, z) t 1D USS HC (t, ) D SS HC (, ) Z 1 With s updated for each time step Z D USS HC (t,, ) With s updated for each time step With s updated for each time step Z 3 With s updated for each time step Figure 1. Guide to worksheet laout for selected partial differential equation solutions MAH 373 Stanle M. Howard 000
3 - Elementar Solutions to Partial Differential Equations (PDQs) 3-3.1 - Partial Derivatives Whenever a dependent variable depends on more than one independent variable, the partial derivative is used to denote how the dependent variable changes with a particular variable. For eample, if (t, ) then the partial derivative of with respect to t ma be approimated as follows: Forward: Backward: Central: t, t, t, t, t, t, + - At time t - + Likewise for the second derivative of with respect to : Central: Forward: Backward: t, t, t, t, t, t, t, t, t, t, t, t, t, t, t, t, t, t, t, t, t, he first order derivatives of with respect to t are as follows: Forward: Backward: Central: t t t tt, t, t t, tt, t tt, tt, t t+t t t-t t-t t t+t South Dakota School of Mines and echnolog Stanle M. Howard 000
3-3 3. - Shorthand Notation for First and Second Partial Derivatives in MAH 373 he previous notation stle for derivative approimations is too cumbersome. A shorter notation is shown below. he position at + is denoted with a plus subscript while the position at - is denoted with a negative subscript. he position at + is denoted with a + subscript. A - is used to denote the position -. he time at t + t is denoted with a superscript prime mark after the dependent variable while the time at t - t is denoted with a prime superscript before the dependent variable. No notation is needed for the location at or the time t. he previous full notation derivatives are now compared with the shorthand notation. ime First Derivatives: Forward: Backward: Central: Full Notation t t t t t t tt t t t t t t tt Shorthand Notation ' t t ' t t ' ' t t (3.1) (3.) (3.3) Position Second Derivatives: Full Notation Central: t, t, t, Forward: t, t, t, Backward: t, t, t, Shorthand Notation (3.4) (3.5) (3.6) 3.3 - Spreadsheet Solution of 1D USS HC Problems he Heat Equation for a 1D USS HC problem with no generation term is (3.7) t Substitution of the finite approimations of the partials gives MAH 373 Stanle M. Howard 000
3 - Elementar Solutions to Partial Differential Equations (PDQs) 3-4 ' (3.8) t Solving for gives ' at (3.9) If the time step, t, is selected such that at 1, then ' 1 (3.10) Create a spreadsheet with one IC and BC s as shown below. he interior s are computed from Eq (3.10). Fill Right and Fill Down in each interior cell labels Initial Condition time BC #1 BC # 3.4 - Spreadsheet Solution of D SS HC Problems he Heat Equation for a D SS HC problem with no generation term is 0 (3.11) Substitution of the finite approimations of the partials and forcing = gives 0 (3.1) South Dakota School of Mines and echnolog Stanle M. Howard 000
3-5 he student should note the use of plus and minus subscripts to denote the direction of Incrementation in the position derivatives. Solving for gives 1 4 his equation ma be used to Rela the solution into the stead state solution b repetitive computation. Be sure to set the spreadsheet to the Iterative mode to avoid circular references errors. See below. (3.13) Create a spreadsheet with two BC s in the -direction and BC s in the -direction as shown below. he interior s are computed from Eq. (3.13) b relaation (repetition). Fill Right and Fill Down in each interior cell -BC s -BC s MAH 373 Stanle M. Howard 000
3 - Elementar Solutions to Partial Differential Equations (PDQs) 3-6 3.5 - Resolving the Circular Reference Error 1) Open the options drawer. ) Select the Formulas menu item. Select Iteration, change from 100 to 1 iteration and select Manual. F9 will be needed to eecute computations on the worksheet. South Dakota School of Mines and echnolog Stanle M. Howard 000
3-7 3.6 - Boundar Conditions here are various boundar conditions encountered in actual heat transfer problems. he most common ones are described here. Fied emperature he fied boundar condition is the simplest of all the boundar conditions. he temperature of the boundar is known. he equation at the surface for the temperature is simpl = fied value (3.14) Gradient Gradient boundar conditions occur when the heat flu at the surface is known. In the simplest case this flu is zero. his condition represents either a perfectl insulated boundar or a boundar at which the geometr requires a minimum or maimum temperature. At the minimum or maimum, the slope of the temperature is zero. herefore, there can be no heat flu. Another wa to identif such a boundar condition is b smmetr. A mirror image temperature profile at a plane is a zero-flu plane. Zero Flu: Zero flu is described as d q = 0 k (3.15) d herefore, d d 0 (3.16) at the zero-flu boundar. his condition can be satisfied b adding a dumm column net to the edge of the zero-flu boundar and making the temperatures in that column equal to the temperatures in the column on the opposite side of the zero boundar flu. If the zero-flu boundar is at =L then the temperature gradient ma be approimated using the central difference approimation d d L L 0 (3.17) herefore, L L (3.18) his is illustrated in the Figure. his is mathematicall the same as setting the boundar to the temperature of the adjacent cell representing the inward temperature one step in from the zero-flu surface. Fied q: A fied-q boundar condition is encountered when the rate of heat loss from a boundar is known. he temperature gradient at the fied-flu boundar of the surface must be matched to the flu through Fourier s Law of Heat Conduction. herefore, for a 1D USS HC problem if the fied-flu boundar is at =L MAH 373 Stanle M. Howard 000
3 - Elementar Solutions to Partial Differential Equations (PDQs) 3-8 Zero flu boundar Dumm column t L- L Make L+ equal to L- L+ Figure. Zero-Flu Boundar Condition in a 1D USS HC Problem. q k d d k L L (3.19) his gives the following equation at the boundar L q L (3.0) k Fied-flu boundar t L- L L- L Figure. Fied-Flu Boundar Condition in a 1D USS HC Problem. Convection: A convection boundar condition is encountered whenever the solid loses heat b convection to a fluid. he rate of heat loss from a boundar is described b the equation q conv h ) (3.1) ( L f South Dakota School of Mines and echnolog Stanle M. Howard 000
3-9 where h = the heat transfer coefficient (which will be known) L = the temperature at the boundar surface f = the temperature of the fluid Since the surface at the boundar has no mass, it cannot store heat. herefore the heat conducted to the surface must be equal to the heat lost from the surface b convection q conv = q cond (3.) substitution of Fourier s Law of Heat and Eq. (3.1) and solving for L gives an equation for the boundar. L c f L 1 c = (3.3) where k c h Radiation: he loss b radiation from a surface is given b the equation q ( ) (3.4) 4 4 rad L S where = the Stephan Boltzmann Constant = 5.510-1 Watts/(K 4 *cm ) = emissivit (0 to 1; 0 for perfectl reflective surfaces, 1 for perfect absorbers) S = the temperature of the surroundings A radiation boundar condition, like the convection condition, requires that the flu to the surface b conduction must equal the flu lost b radiation heat transfer. herefore, q rad = q cond (3.5) substitution of Fourier s Law of Heat and Eq. (3.4) and solving for L gives a transcendental equation for the boundar temperature L 4 4 L L ( L S ) k (3.6) Since this equation is transcendental, no eplicit epression can be written for L. his complicates solving problems with radiation boundar conditions b requiring a numerical convergence (iteration) routine to solve for the boundar temperature. On a spreadsheet, this requires the use of a macro function that emplos a root finding method. 3.7 - Spreadsheet Solution of a D USS HC Problem he governing equation for a D USS HC problem with no generation is (3.7) t MAH 373 Stanle M. Howard 000
3 - Elementar Solutions to Partial Differential Equations (PDQs) 3-10 Substituting in the central difference approimations for the second order partial terms and the forward difference approimation for the first order partial and solving for the forward gives ' t (3.8) he student should note the use plus and minus subscripts to denote the direction of incrementation in the position derivatives. t 1 If t is set at a value such that, then Eq. (3.8) becomes 4 ' (3.9) 4 It is ver important to recognize that this looks like the same result as for a D SS HC problem but that it is different in that the left side is, not simpl. he calculation of a new temperature using Eq.(3.9) requires that all new temperatures are computed from ALL old temperatures. In the D SS HC solution the old and new temperatures appearing during relaation had nothing to do with time. herefore, to solve a D USS HC problem using Eq. (3.9) will require a panel of old s from which all new s are computed. Once all the new s are computed for all positions in and, those s become the old s another panel of new s is computed. If a new panel is used for each time step, then a ver large number of panels would be needed to complete a solution. he need for man panels can be reduced to just two panels if, rather than using a third panel for the second computation, the first panel is used to hold the results for the second time step. he second panel could then be used to hold the results for the third time step and so on for as man time steps as desired. Figure 3 summarizes the calculation scheme. Panel 1 Panel Panel 1 Holds s @ t=0, t, 4t, 6t, etc Panel Holds s @ t=t, 3t, 5t, etc Figure 3. Calculation Scheme oggle and s If one chooses to use a worksheet to solve a D USS HC problem, some means of controlling when the left or right panel computes is required. A variable s is used to reset the spreadsheet in case something goes wrong and the cells fill up with REF or other tet indicating a problem. With a means of resetting the cell value, everthing must be retped and even then there is no guarantee the same problem will necessitate another retping: a ver discouraging prospect. Of course if one chooses to write code (VBA, MALAB), no such resetting procedure is needed if an error occurs. Whenever s is set to 0, the spreadsheet cell values return to the initial temperatures as defined b the initial condition and the value of L is reset to 0. When s is set to a value other than 0, L will alternate South Dakota School of Mines and echnolog Stanle M. Howard 000
3-11 between 0 and 1 each calculation ccle. his is accomplished with the following statement defining L (that is, the name of the variable L is created using the create variable menu command) to be =IF(s, IF (L, 0, 1), 0) (3.30) he value of L is used to turn on and off calculations on Panel 1 (left) and (right). A tpical statement for Panel 1, cell D10 would be =IF(s, IF(L,(AD9+AD11+AC10+AE10)/4, D10), IC) (3.31) assuming that Panel is one alphabetic ccle to the right of Panel 1. A similar statement is used for cell AD10. =IF(s, IF(L,AD10,(D9+D11+C10+E10)/4), IC) (3.3) A time counter must be included. he cell holding the equation for the created name for time, t, would have the form =IF(s, t+t, 0). (3.33) Boundar Conditions he edges of each panel are determined b the four Boundar Conditions. 3.8 When hermal Conductivit is a Linear Function of emperature he derivation of the partial differential equation for one-dimensional unstead state heat transfer begins with the one-dimensional heat balance. q Cp t (3.34) Substituting Fourier s Law for q gives d k d Cp t (3.35) Making k a linear function of temperature gives d ab d Cp t (3.36) which becomes d d d d t ab b Cp (3.37) MAH 373 Stanle M. Howard 000
3 - Elementar Solutions to Partial Differential Equations (PDQs) 3-1 Estimating the derivatives using the central differences for the partials with respect to position and the forward difference for the time derivative gives ab b Cp t (3.38) Solving for provides the basis for the approimate numerical solution. ab bt t Cp Cp (3.39) South Dakota School of Mines and echnolog Stanle M. Howard 000