On the Transient Atomic/Heat Diffusion in Cylinders and Spheres with Phase Change: A Method to Derive Closed-Form Solutions

<div>Comparison of concentration profiles during inward single-phase solidification of a slab, a cylinder, and a sphere as a function of dimensionless position <svg height="14.2236pt" id="M235" style="vertical-align:-4.93527pt" version="1.1" viewbox="-0.0498162 -9.28833 40.3752 14.2236" width="40.3752pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.584,3.132)"><path d="M435 683C435 695 433 700 417 700C380 700 311 676 267 649C217 655 198 677 198 698C198 713 202 725 205 731L185 739C156 728 122 706 122 679C122 646 165 618 221 611C170 578 115 533 115 461C115 416 145 377 177 365C103 329 24 261 24 163C24 72 81 34 155 14C181 7 201 3 222 -1C284 -14 298 -34 298 -59C298 -97 254 -151 235 -171L262 -194C285 -179 378 -110 378 -28C378 14 343 58 273 70C248 74 219 80 190 89C140 104 109 134 109 190C109 259 165 325 225 350C251 343 285 340 318 340C334 340 376 381 376 403C376 416 372 420 355 420C323 420 284 410 258 403C221 409 194 444 194 490C194 540 236 586 274 608C317 608 364 613 388 623C408 631 435 653 435 683Z"></path></g><g transform="matrix(.013,0,0,-0.013,17.275,0)"><path d="M368 703H309L44 -163H104L368 703Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.636,0)"><path d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,30.722,3.132)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z"></path></g><g transform="matrix(.013,0,0,-0.013,35.668,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg> at 70% of <span class="nowrap"><svg height="11.927pt" id="M236" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 13.1624 11.927" width="13.1624pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.086,3.132)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z"></path></g></svg>.</span></div>

International Journal of Mathematics and Mathematical Sciences

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Figure 12

Figure 12: On the Transient Atomic/Heat Diffusion in Cylinders and Spheres with Phase Change: A Method to Derive Closed-Form Solutions 