ECAR 131 Verification Problems

Overview

ECAR 131 is an INL internal report describing a set of simulations used to verify the quality of Abaqus solutions. The problems described here are taken from ECAR 131. The analytic solutions for these problems come from Incropera and DeWitt (2001).

An accuracy tolerance of 3% is used in ECAR 131.

Nomenclature

Table 1: Symbols and their meanings.

Symbol	Definition
$var element = document.getElementById("moose-equation-44450e73-96fa-4105-a4e2-4078161875a5");katex.render("k", element, {displayMode:false,throwOnError:false});$	thermal conductivity
$var element = document.getElementById("moose-equation-8d08d904-15f3-4338-bdd1-01c3a5ea1b3b");katex.render("c", element, {displayMode:false,throwOnError:false});$	specific heat
$var element = document.getElementById("moose-equation-21cdaaf3-8a3b-4d9c-91a3-7f94e53e6657");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$	density
$var element = document.getElementById("moose-equation-d41d3455-4902-4619-9e65-32ddaf403823");katex.render("h", element, {displayMode:false,throwOnError:false});$	heat transfer coefficient
$var element = document.getElementById("moose-equation-bd6f0373-83b3-42ca-a39c-8a6ac5ffb2b5");katex.render("L", element, {displayMode:false,throwOnError:false});$	length
$var element = document.getElementById("moose-equation-1f7dc33b-a910-4f7b-9ced-5a5e93ebdd2f");katex.render("R", element, {displayMode:false,throwOnError:false});$	radius
$var element = document.getElementById("moose-equation-df365fa6-72b1-4c70-8fbd-932f9d062c7f");katex.render("x", element, {displayMode:false,throwOnError:false});$	Cartesian coordinate
$var element = document.getElementById("moose-equation-5c99d347-c59f-4f2f-b8d9-83c24c7ebcd1");katex.render("r", element, {displayMode:false,throwOnError:false});$	radial coordinate
$var element = document.getElementById("moose-equation-12463315-471f-4211-989e-3f587ee49cee");katex.render("T", element, {displayMode:false,throwOnError:false});$	temperature
$var element = document.getElementById("moose-equation-d2dad9d4-2998-4ab1-b58a-8d42742dc0e8");katex.render("q", element, {displayMode:false,throwOnError:false});$	heat flux
$var element = document.getElementById("moose-equation-68c6a3a3-46f6-41f4-b8c3-9e2e2c621e37");katex.render("m", element, {displayMode:false,throwOnError:false});$	mass flow rate
$var element = document.getElementById("moose-equation-bfbf2da9-85cf-4eac-9929-d56b22c116b1");katex.render("t", element, {displayMode:false,throwOnError:false});$	time
$var element = document.getElementById("moose-equation-072dfe3b-bda2-47ce-bc11-b9a4cb675ee2");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$	emissivity
$var element = document.getElementById("moose-equation-65c8d602-e92f-4c83-b7dd-bee3ba3ab07f");katex.render("\\sigma", element, {displayMode:false,throwOnError:false});$	Stefan-Boltzmann constant
$var element = document.getElementById("moose-equation-262ea738-2edc-4b34-b895-9fb9c5b80b8e");katex.render("\\delta", element, {displayMode:false,throwOnError:false});$	relative error
$var element = document.getElementById("moose-equation-a9d7f946-6468-4502-8c46-33766dab5121");katex.render("\\alpha=k/\\rho/c", element, {displayMode:false,throwOnError:false});$	thermal diffusivity
$var element = document.getElementById("moose-equation-96a677b2-5d4d-43fb-8b1f-9dfe9f6ac1a1");katex.render("B=h L / k", element, {displayMode:false,throwOnError:false});$	Biot number
$var element = document.getElementById("moose-equation-e4664e84-8868-4215-a702-525c9d2b0e55");katex.render("F=k t/\\rho/c/L^2", element, {displayMode:false,throwOnError:false});$	Fourier number
$var element = document.getElementById("moose-equation-1639d6af-3a3b-4df4-ba8b-70c594ca0e8f");katex.render("S=h/m/c", element, {displayMode:false,throwOnError:false});$	Stanton number

Plane Wall with Convection (Problem #1)

This problem is a wall with one surface insulated and the other surface having a convection boundary condition. The insulated surface is at $var element = document.getElementById("moose-equation-ba9acad7-3ba7-4ea3-9677-6a841ea28bdc");katex.render("x=0", element, {displayMode:false,throwOnError:false});$ , and the surface with the convection boundary condition is at $var element = document.getElementById("moose-equation-f6276739-417e-4ed0-aaa3-9816510022e5");katex.render("x=L", element, {displayMode:false,throwOnError:false});$ . The parameters for this problem are in Table 2.

Table 2: Parameters for plane wall problem.

Symbol	Value
$var element = document.getElementById("moose-equation-6404a886-6340-4c4c-8463-e97949640660");katex.render("k", element, {displayMode:false,throwOnError:false});$	0.45 W/in/K
$var element = document.getElementById("moose-equation-829ea576-ed7a-489e-868f-f6565f2f19ff");katex.render("c", element, {displayMode:false,throwOnError:false});$	4.2E5 J/kg/K
$var element = document.getElementById("moose-equation-d19ba11d-6074-4072-8216-9fb09ba8c15a");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$	2.7E-5 kg/in $var element = document.getElementById("moose-equation-2d850212-91fd-4641-8730-752fd6b898ed");katex.render("^3", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-880fd9d3-ddd4-44ec-aa4b-73baea9a77b6");katex.render("h", element, {displayMode:false,throwOnError:false});$	0.375 W/in $var element = document.getElementById("moose-equation-510c419e-15f3-4ae2-8c48-c174e15d3b66");katex.render("^2", element, {displayMode:false,throwOnError:false});$ /K
$var element = document.getElementById("moose-equation-39d84538-af42-4e3e-bebe-19216b0d54a4");katex.render("L", element, {displayMode:false,throwOnError:false});$	1.2 in
$var element = document.getElementById("moose-equation-712381bb-f15a-4f65-9317-723df38c9641");katex.render("T_a", element, {displayMode:false,throwOnError:false});$	4 $var element = document.getElementById("moose-equation-456767aa-80fb-4356-8771-beda1cf1cb39");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C (fluid temperature)
$var element = document.getElementById("moose-equation-75eacef0-07d1-4d55-bd04-43a5f08fdea0");katex.render("T_i", element, {displayMode:false,throwOnError:false});$	60 $var element = document.getElementById("moose-equation-be7ea745-07c3-47ea-8f77-a70831862a91");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C
$var element = document.getElementById("moose-equation-1062caa4-bea4-4c4d-a8d3-d45c774f6cd9");katex.render("\\zeta", element, {displayMode:false,throwOnError:false});$	0.8603 rad
$var element = document.getElementById("moose-equation-74e89a58-1247-4d3f-a05f-73124e71fd31");katex.render("C_1", element, {displayMode:false,throwOnError:false});$	1.1191

The analytic solution is given as $var element = document.getElementById("moose-equation-93ad22de-20b6-4a6f-829a-b8fa48f5f3fc");katex.render(" T=T_a+(T_i-T_a)C_1\\text{exp}\\left(-\\zeta^2F\\right)\\text{cos}\\left(\\zeta\\frac{x}{L}\\right)", element, {displayMode:true,throwOnError:false});$

At $var element = document.getElementById("moose-equation-e17fb067-81ed-40ac-b293-52a37d397fef");katex.render("t", element, {displayMode:false,throwOnError:false});$ = 90 seconds, the temperature at the insulated wall is 13.997 $var element = document.getElementById("moose-equation-1122fce4-b8f5-4bc3-bbba-1591407264c8");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C, and the temperature at the other wall is 10.52 $var element = document.getElementById("moose-equation-c8e450b4-f309-4785-9b69-144e8e243c7c");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C.

Table 3: Results for plane wall with convection.

Wall	Analytic	Bison	Error
Insulated	13.997 $var element = document.getElementById("moose-equation-eea4781e-eef0-4374-90d5-29715fe86532");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	14.151 $var element = document.getElementById("moose-equation-520cab57-e881-464b-a464-6ffc71f3acd0");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	1.10%
Convection	10.52 $var element = document.getElementById("moose-equation-82bb69c7-13d7-43f6-af50-c0b3b8ba626f");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	10.620 $var element = document.getElementById("moose-equation-94906da5-8572-4b4f-afdd-1a6e02786647");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0.950%

Plane Wall with Convection, 2D (Problem #2)

The problem is the same as the previous one but in 2D instead of 3D.

At $var element = document.getElementById("moose-equation-b1aa0541-56f2-4cc4-977c-7aeddffc15fd");katex.render("t", element, {displayMode:false,throwOnError:false});$ = 90 seconds, the temperature at the insulated wall is 13.997 $var element = document.getElementById("moose-equation-5c031385-1389-4faf-9575-1da049a722f5");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C, and the temperature at the other wall is 10.52 $var element = document.getElementById("moose-equation-f0788eba-c25c-4d4c-a2cc-dc304d68b5eb");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C.

Table 4: Results for 2D plane wall with convection.

Wall	Analytic	Bison	Error
Insulated	13.997 $var element = document.getElementById("moose-equation-42fcdef4-115f-4c27-8c7e-d7dc16f427f9");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	14.151 $var element = document.getElementById("moose-equation-fd60ad0a-4012-4bdd-94c1-4bba70fdab3e");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	1.10%
Convection	10.52 $var element = document.getElementById("moose-equation-846854a1-16b1-40c7-adf8-b7749fdecb6c");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	10.620 $var element = document.getElementById("moose-equation-6f6dc2b6-5538-4cf2-9f7b-5c6215174f08");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0.950%

Cylinder with Convection (Problem #3)

This problem is a cylinder with a convection boundary condition on the radial surface.

The parameters are the same as for Plane Wall with Convection (Problem #1) except that $var element = document.getElementById("moose-equation-40a35f57-5ffd-4582-b01c-526a25f1fa49");katex.render("L", element, {displayMode:false,throwOnError:false});$ is replaced with $var element = document.getElementById("moose-equation-885c3ded-aaa5-4a32-8a78-ae608154cfe8");katex.render("R", element, {displayMode:false,throwOnError:false});$ = 0.6 in.

The analytic solution is given as $var element = document.getElementById("moose-equation-0f1e9338-e712-4178-b7a0-0709aa5b927d");katex.render(" T=T_a+(T_i-T_a)C_1\\text{exp}\\left(-\\zeta^2F\\right)J_0\\left(\\zeta\\frac{x}{R}\\right)", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-4267e1f1-1e89-4184-9f03-93d15d4778bd");katex.render("J_0", element, {displayMode:false,throwOnError:false});$ is the Bessel function of the first kind, $var element = document.getElementById("moose-equation-934d54cb-32ce-4e3b-876d-e30c29082804");katex.render("J_n", element, {displayMode:false,throwOnError:false});$ with $var element = document.getElementById("moose-equation-aa0a9831-f736-4c42-8ed2-d9d6ba8705ec");katex.render("n", element, {displayMode:false,throwOnError:false});$ = 0.

At $var element = document.getElementById("moose-equation-285ef02f-fa3b-41fe-8de5-733d9e35874f");katex.render("t", element, {displayMode:false,throwOnError:false});$ = 18 seconds, the temperature at the wall is 12.521 $var element = document.getElementById("moose-equation-fe920f2f-02a8-4b22-97f8-5609f23bc427");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C, and the temperature at the center of the cylinder is 14.777 $var element = document.getElementById("moose-equation-46f532b8-e686-4b94-bf49-c39d50753cd5");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C.

Table 5: Results for cylinder with convection.

Location	Analytic	Bison	Error
Wall	12.521 $var element = document.getElementById("moose-equation-36733739-b10f-4987-9472-f97b23d3f715");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	12.753 $var element = document.getElementById("moose-equation-a1243c92-884f-4e44-a53b-1b58705de6d4");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	1.85%
Center	14.777 $var element = document.getElementById("moose-equation-b6643ecb-c26e-45a3-93b6-475d04946ffe");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	15.074 $var element = document.getElementById("moose-equation-e93274a7-f88a-41a0-b1b7-dcfdcb09e23a");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	2.01%

Plane Wall with Internal Heat Generation (Problem #4)

This problem is similar to Plane Wall with Convection (Problem #1). However, instead of a convection boundary condition at one wall, the temperature at that wall is fixed, $var element = document.getElementById("moose-equation-620a58fd-d770-45d4-b4d4-1e931c6c696b");katex.render("T_w", element, {displayMode:false,throwOnError:false});$ = 4 $var element = document.getElementById("moose-equation-fc14ec29-adf2-4de8-bbbc-80d760f0e13b");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C. Also, the wall has internal heat generation $var element = document.getElementById("moose-equation-7dc5f844-b13f-450c-a42d-f3d0b8dbde23");katex.render("Q", element, {displayMode:false,throwOnError:false});$ = 6.1 W/in $var element = document.getElementById("moose-equation-b45a9083-1450-45ca-9939-81fc2fc12f2d");katex.render("^3", element, {displayMode:false,throwOnError:false});$ .

The analytic solution is given as $var element = document.getElementById("moose-equation-27b62f13-553c-4ea3-90f7-7c3f771763e6");katex.render(" T=T_w+\\frac{QL^2}{2k}\\left(1-x^2\\right)", element, {displayMode:true,throwOnError:false});$

Table 6: Results for plane wall with heat generation.

Coordinate	Analytic	Bison	Error
0.0	13.76 $var element = document.getElementById("moose-equation-41a693e5-5fa9-45a5-88f0-7f42c66931ae");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	13.76 $var element = document.getElementById("moose-equation-49b48d81-faca-4d59-9edd-2648187ca6b1");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0%
0.3	13.15 $var element = document.getElementById("moose-equation-eca297dd-7b0f-4be1-960d-3cb529c0bab2");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	13.15 $var element = document.getElementById("moose-equation-0c068f4a-9517-4093-98d6-937c6e9f5540");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0%
0.6	11.32 $var element = document.getElementById("moose-equation-e89988b7-9e06-48af-ad17-d22bc38e843c");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	11.32 $var element = document.getElementById("moose-equation-9d752412-480f-4ddb-b6a9-1215c124cd4d");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0%
0.9	8.27 $var element = document.getElementById("moose-equation-23985b6d-0927-4e81-8bd5-daf572431b87");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	8.27 $var element = document.getElementById("moose-equation-8d067222-e15a-44be-b9ea-43cce95c0832");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0%
1.2	4.00 $var element = document.getElementById("moose-equation-c3d6c5fb-6984-43a8-bac5-50e325ecd2fe");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	4.00 $var element = document.getElementById("moose-equation-1dedce8a-d13e-431b-87cd-fe43d2af0148");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0%

Plane Wall with Internal Heat Generation, 2D (Problem #5)

This problem is the same as the previous one but in 2D instead of 3D.

The analytic solution is given as $var element = document.getElementById("moose-equation-3abaa39e-0431-402f-a299-b405618b4976");katex.render(" T=T_w+\\frac{QL^2}{2k}\\left(1-x^2\\right)", element, {displayMode:true,throwOnError:false});$

Table 7: Results for plane wall with heat generation.

Coordinate	Analytic	Bison	Error
0.0	13.76 $var element = document.getElementById("moose-equation-c6752bb7-14c4-4ea8-9a8b-caea9a4dffd4");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	13.76 $var element = document.getElementById("moose-equation-fe4e860b-e5eb-4b73-94bc-5342fc5d1cd2");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0%
0.3	13.15 $var element = document.getElementById("moose-equation-99650ff0-4324-446f-acd7-6a86b492e16e");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	13.15 $var element = document.getElementById("moose-equation-f92e3f5a-8b2f-4a7b-b655-8aee69079747");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0%
0.6	11.32 $var element = document.getElementById("moose-equation-7e8e2110-98d7-42d9-ae89-f24811da8c61");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	11.32 $var element = document.getElementById("moose-equation-9a74c821-98c0-482d-8cb0-0abbdac5ef46");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0%
0.9	8.27 $var element = document.getElementById("moose-equation-d984fc79-e675-44e1-9beb-21f6a3123bee");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	8.27 $var element = document.getElementById("moose-equation-9f04391a-a6ef-47ff-a87f-8b00a8e09bb7");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0%
1.2	4.00 $var element = document.getElementById("moose-equation-8140f02b-ebc1-46df-8d99-c90fb03c4d6d");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	4.00 $var element = document.getElementById("moose-equation-943fd91a-e145-4947-978e-8884bc752600");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0%

Semi-infinite Plate with Heat Flux (Problem #6)

The parameters are the same as for Plane Wall with Convection (Problem #1) except that the heat flux at the surface is given as $var element = document.getElementById("moose-equation-78978706-7830-4258-a97f-cd9e1947dee8");katex.render("q", element, {displayMode:false,throwOnError:false});$ = 7.3 W/m $var element = document.getElementById("moose-equation-dc0f7454-4722-44e8-b3bf-582449e3295b");katex.render("^3", element, {displayMode:false,throwOnError:false});$ .

The analytic solution is given as $var element = document.getElementById("moose-equation-65a358b8-76a1-4bed-a246-898557cf1219");katex.render(" T=T_i+\\frac{2q}{k}\\sqrt{\\frac{\\alpha t}{\\pi}}\\text{exp}\\left(\\frac{-x^2}{4\\alpha t}\\right)-\\frac{qx}{k}\\text{erfc}\\left(\\frac{x}{2\\sqrt{\\alpha t}}\\right)", element, {displayMode:true,throwOnError:false});$

At $var element = document.getElementById("moose-equation-150e994d-a8ee-47ea-b728-a5c7ab6d968d");katex.render("t", element, {displayMode:false,throwOnError:false});$ = 90 seconds, the temperature at the surface is 94.593 $var element = document.getElementById("moose-equation-026b663b-1fc8-4747-9a8d-9f133e9261e9");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C, and the temperature at a depth of 1 in is 80.764 $var element = document.getElementById("moose-equation-002442eb-d5a3-4f85-ab6e-2b9a40f1792e");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C.

Table 8: Results for semi-infinite plate with heat flux.

Location	Analytic	Bison	Error
Surface	94.593 $var element = document.getElementById("moose-equation-de31bd48-1e4b-46bf-ae53-1596d73bd2eb");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	94.497 $var element = document.getElementById("moose-equation-1d3417b6-8ae7-45c1-bd72-648fe6a30d75");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	-0.101%
Depth 1 in.	80.764 $var element = document.getElementById("moose-equation-19f36db0-c5cf-4e10-ab16-34a5dc42b457");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	80.729 $var element = document.getElementById("moose-equation-9b161ee0-79ef-452f-8abe-8fdf52d98997");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	-0.0431%

Radiation between Parallel Plates (Problem #7)

The problem as defined in ECAR 131 uses shell elements. We instead use 2D elements.

This problem is one of radiation heat transfer between parallel plates. The temperature at one boundary is 278 K, and at the other boundary the temperature is 333 K. The emissivities of the two boundaries are 0.8 and 0.9. The value of the Stefan-Boltzmann constant, $var element = document.getElementById("moose-equation-a9c1ee21-6a4c-4086-a7f8-6c9c81fce4e0");katex.render("\\sigma", element, {displayMode:false,throwOnError:false});$ , is 5.670374e-8 W/m $var element = document.getElementById("moose-equation-8fb71707-0bf9-4159-b5e3-a3e65095ccf8");katex.render("^2", element, {displayMode:false,throwOnError:false});$ /K $var element = document.getElementById("moose-equation-e911fb1a-80a3-47a2-9a76-86b05aab9853");katex.render("^4", element, {displayMode:false,throwOnError:false});$ .

The analytic solution for the flux is given as $var element = document.getElementById("moose-equation-d7846807-fc17-4d01-a022-d71fd67e2c7a");katex.render(" q_1 = \\frac{\\sigma(T_1^4-T_2^4)}{\\frac{1}{\\epsilon_1}+\\frac{1}{\\epsilon_2}-1}", element, {displayMode:true,throwOnError:false});$

Table 9: Radiation flux between parallel plates.

Analytic	Bison	Error
-263.44 W/m $var element = document.getElementById("moose-equation-bec29206-a285-433e-87a2-7b4d1f6dc233");katex.render("^2", element, {displayMode:false,throwOnError:false});$	-261.25 W/m $var element = document.getElementById("moose-equation-7a5b5cbe-4916-489c-ae61-8092a836cfc4");katex.render("^2", element, {displayMode:false,throwOnError:false});$	-0.83%

Radiation between Concentric Cylinders (Problem #8)

This problem is conceptually similar to the previous problem. We use 2D elements instead of shells, as were used in ECAR 131.

The inner radius, $var element = document.getElementById("moose-equation-02cd0166-f566-4329-8f8b-ebc8c34061ba");katex.render("R_1", element, {displayMode:false,throwOnError:false});$ , is 0.5 in. The outer radius, $var element = document.getElementById("moose-equation-7c03a3ad-5fae-4bd3-8a5d-573ce2102dc8");katex.render("R_2", element, {displayMode:false,throwOnError:false});$ , is 0.525 in.

The analytic solution for the flux $var element = document.getElementById("moose-equation-7dfb2694-1159-4ba0-aca1-49de3bc4b8a2");katex.render("q_1", element, {displayMode:false,throwOnError:false});$ is $var element = document.getElementById("moose-equation-74a82c3f-58f7-46b2-bca5-d5846fe71795");katex.render(" q_1 = \\frac{\\sigma(T_1^4-T_2^4)}{\\frac{1}{\\epsilon_1}+\\frac{1-\\epsilon_2}{\\epsilon_2}\\frac{R_1}{R_2}}", element, {displayMode:true,throwOnError:false});$

The analytic solution for the flux $var element = document.getElementById("moose-equation-f1ba7651-17d9-4e8d-83a0-b0e8d9fe9a33");katex.render("q_2", element, {displayMode:false,throwOnError:false});$ is $var element = document.getElementById("moose-equation-03e5c9e8-647b-4893-b935-360f51510c3f");katex.render(" q_2 = -q_1\\frac{R_1}{R_2}", element, {displayMode:true,throwOnError:false});$

Table 10: Radiation flux between concentric cylinders.

Flux	Analytic	Bison	Error
$var element = document.getElementById("moose-equation-3a62c4cd-6564-4567-b218-a71d55a94b83");katex.render("q_1", element, {displayMode:false,throwOnError:false});$	-264.47 W/m $var element = document.getElementById("moose-equation-92f989ba-9539-4dff-9949-bb87ac2296a8");katex.render("^2", element, {displayMode:false,throwOnError:false});$	-264.37 W/m $var element = document.getElementById("moose-equation-56ce51aa-889b-4aae-b084-4a892ee137ee");katex.render("^2", element, {displayMode:false,throwOnError:false});$	-0.037%
$var element = document.getElementById("moose-equation-284f70ed-74c2-4098-a915-c2bd344b1532");katex.render("q_2", element, {displayMode:false,throwOnError:false});$	251.87 W/m $var element = document.getElementById("moose-equation-033d433f-d9ab-4669-844f-d0770b2f583d");katex.render("^2", element, {displayMode:false,throwOnError:false});$	251.78 W/m $var element = document.getElementById("moose-equation-3357ea1e-82f3-44ad-8635-b15fc5e67c21");katex.render("^2", element, {displayMode:false,throwOnError:false});$	-0.036%

Convective Heat Transfer to a Fluid Flowing in a Circular Tube (Problem #9)

This problem is heat transfer to a fluid in a circular tube. The temperature at the surface of the tube is fixed, $var element = document.getElementById("moose-equation-c798961f-01c0-4e5d-86dc-8b671a3fd6b8");katex.render("T_s", element, {displayMode:false,throwOnError:false});$ = 50 $var element = document.getElementById("moose-equation-f93918ce-99a5-420c-b9a8-21b8e66a8be6");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C. The parameters for this problem are in Table 11.

Table 11: Parameters for convective heat transfer to a fluid.

Symbol	Value
$var element = document.getElementById("moose-equation-f7c2f7d6-602e-4fa1-ac24-b4f6da2b94dd");katex.render("h", element, {displayMode:false,throwOnError:false});$	189 W/in $var element = document.getElementById("moose-equation-ea04edb2-7e61-4c74-8f08-b79ab9adf673");katex.render("^2", element, {displayMode:false,throwOnError:false});$ /K
$var element = document.getElementById("moose-equation-e3a0ae63-d47b-4012-bd05-cdde75b05c4f");katex.render("c", element, {displayMode:false,throwOnError:false});$	4.2E5 J/kg/K
$var element = document.getElementById("moose-equation-a4f2931c-d5ba-485a-9da9-b87c156c57cf");katex.render("R_{tube}", element, {displayMode:false,throwOnError:false});$	29 in
$var element = document.getElementById("moose-equation-040afbcf-ee76-46ec-b2ed-a0556802f3da");katex.render("m", element, {displayMode:false,throwOnError:false});$	4.5E-4 kg/in $var element = document.getElementById("moose-equation-382e1e07-50bb-45e2-be42-e07f541b2a3a");katex.render("^2", element, {displayMode:false,throwOnError:false});$ /s
$var element = document.getElementById("moose-equation-0c6e0a14-fd94-44f2-8a3d-d72bf9610062");katex.render("L", element, {displayMode:false,throwOnError:false});$	120 in
$var element = document.getElementById("moose-equation-251486bb-9f33-43ec-9041-c16f0ea96935");katex.render("T_i", element, {displayMode:false,throwOnError:false});$	55.55 $var element = document.getElementById("moose-equation-dae26783-1ae5-4280-9ae3-def3c30102ed");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C
$var element = document.getElementById("moose-equation-8b6f0eab-d55a-4faa-a1da-8178f13e674c");katex.render("S", element, {displayMode:false,throwOnError:false});$	1.0
$var element = document.getElementById("moose-equation-365337cd-1772-449e-adf2-2688cbc32729");katex.render("y", element, {displayMode:false,throwOnError:false});$	2 $var element = document.getElementById("moose-equation-8d9a23d9-705a-4657-a14a-f3823b8af213");katex.render("S", element, {displayMode:false,throwOnError:false});$ / $var element = document.getElementById("moose-equation-23c65bb3-9799-4656-ae00-b3ad4d4ac045");katex.render("R_{tube}x", element, {displayMode:false,throwOnError:false});$

The analytic solution for the average fluid temperature is given as $var element = document.getElementById("moose-equation-03caca52-a8d3-4020-96f0-4d81d9dca3cd");katex.render(" T=T_s+(T_i-T_s)\\text{exp}(-y)", element, {displayMode:true,throwOnError:false});$

Table 12: Results for convective heat transfer to a moving fluid.

Position	Analytic	Bison	Error
0.0	55.55 $var element = document.getElementById("moose-equation-88d0e7b8-1c48-4e9b-98cd-1901feb6c12c");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	55.55 $var element = document.getElementById("moose-equation-0bfb436d-16a3-423c-91ba-ee9db1b4352d");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0.00%
6.0	53.669 $var element = document.getElementById("moose-equation-1fdfa2d7-2c0a-4b2f-9b22-66d88998368f");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	54.085 $var element = document.getElementById("moose-equation-49f7c7a5-03a6-48b3-8c30-c32c69c77cf5");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0.78%
12.0	52.426 $var element = document.getElementById("moose-equation-d20cdada-5fc1-4bd8-b9a5-ff9a105261c2");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	52.815 $var element = document.getElementById("moose-equation-b52da673-063f-4ed5-91dd-58cc4d739e3d");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0.74%
18.0	51.604 $var element = document.getElementById("moose-equation-c518918d-ae41-4828-a7a7-b295e5240c06");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	51.848 $var element = document.getElementById("moose-equation-a6933656-2a4e-41ce-aef9-cf0cb1755fd4");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0.47%
24.0	51.06 $var element = document.getElementById("moose-equation-268b6ff9-8e84-40db-bc2b-e5201506b11d");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	51.172 $var element = document.getElementById("moose-equation-283e07bf-7790-4a21-afda-3cae4a3fa8d6");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0.22%
30.0	50.701 $var element = document.getElementById("moose-equation-466989d9-cbd9-4fa2-8f8b-8dc9c2de85d3");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	50.730 $var element = document.getElementById("moose-equation-bfbb41d6-dee0-467a-8d9f-d8a598718f4e");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0.057%
42.0	50.306 $var element = document.getElementById("moose-equation-7f2a5ba1-f6ca-4d2e-90c8-1a00ca85a98f");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	50.275 $var element = document.getElementById("moose-equation-49f6864f-5f5f-4b89-b9b1-f34f0feac439");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0.063%
60.0	50.089 $var element = document.getElementById("moose-equation-ff2d29a4-3038-438a-befe-4885cee6edd3");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	50.062 $var element = document.getElementById("moose-equation-06c056ae-5226-42e0-be27-44ce7719d9a4");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0.053%
90.0	50.011 $var element = document.getElementById("moose-equation-8341dfb3-a543-4f54-8243-fc0745c31dfa");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	50.005 $var element = document.getElementById("moose-equation-1c0ed92e-a25d-4fec-b0e1-c9c3567efeef");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0.012%
120.0	50.001 $var element = document.getElementById("moose-equation-05ca17dd-0ce9-4369-9d17-e3a0168a9b26");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	50.001 $var element = document.getElementById("moose-equation-dbaaee4f-45f9-4743-8faa-ef1a5d7dc1a0");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C	0.0011%

Plane Wall with Radiative Boundary (Problem #10)

This problem is a wall with one surface insulated and the other surface with a radiative boundary condition. The insulated surface is at $var element = document.getElementById("moose-equation-2117cb46-defd-489e-b843-091950b2d3ec");katex.render("x=0", element, {displayMode:false,throwOnError:false});$ , and the surface with the convection boundary condition is at $var element = document.getElementById("moose-equation-a32a597d-3883-4d34-8a35-77dc80b19edd");katex.render("x=L", element, {displayMode:false,throwOnError:false});$ . The parameters for this problem are in Table 13.

Table 13: Parameters for plane wall problem with radiative boundary condition.

Symbol	Value
$var element = document.getElementById("moose-equation-637e4fa8-969f-40fb-b755-e058ec0da2c8");katex.render("k", element, {displayMode:false,throwOnError:false});$	45 W/in/K
$var element = document.getElementById("moose-equation-483fa220-c847-4744-9b4b-f50cc872c3f0");katex.render("c", element, {displayMode:false,throwOnError:false});$	4.2E3 J/kg/K
$var element = document.getElementById("moose-equation-65179a34-757f-4090-aab5-4a68ddceba86");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$	2.7E-5 kg/in $var element = document.getElementById("moose-equation-7ee9f88c-f64e-4f37-9599-848c83e023bf");katex.render("^3", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-c4d4da2f-028b-4e99-857b-3f977b627830");katex.render("\\epsilon", element, {displayMode:false,throwOnError:false});$	1.0
$var element = document.getElementById("moose-equation-e95f3bd7-69a7-45ab-94de-25b4c1956b4e");katex.render("L", element, {displayMode:false,throwOnError:false});$	1.2 in
$var element = document.getElementById("moose-equation-cb3ad990-dcbc-403d-b54f-beda04b45f7e");katex.render("T_a", element, {displayMode:false,throwOnError:false});$	0 K (ambient temperature)
$var element = document.getElementById("moose-equation-b356a68f-5d07-443f-9fab-ef01dfd84d91");katex.render("T_i", element, {displayMode:false,throwOnError:false});$	333.15 K
$var element = document.getElementById("moose-equation-9ba3070e-08b5-47d0-88df-c8161dbff9f3");katex.render("\\sigma", element, {displayMode:false,throwOnError:false});$	3.658E-11 W/in $var element = document.getElementById("moose-equation-bf5fac6e-b749-4ac2-98cf-6ed2e21b7c7e");katex.render("^2", element, {displayMode:false,throwOnError:false});$ /K $var element = document.getElementById("moose-equation-91144208-e2a4-45d3-a0ab-62dcab105681");katex.render("^4", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-8b5c1a9d-813b-4008-a7e1-f45c94628616");katex.render("C_1", element, {displayMode:false,throwOnError:false});$	1.1191

The analytic solution is given as $var element = document.getElementById("moose-equation-d7473e5e-2ba5-46bd-aabf-04b671683ac8");katex.render(" T=T_i\\left( 1+3\\frac{\\sigma\\epsilon T_i^3 t}{\\rho c L}\\right)^\\frac{-1}{3}", element, {displayMode:true,throwOnError:false});$

At $var element = document.getElementById("moose-equation-2978f620-e96d-414c-a2f3-17bd9562ebcb");katex.render("t", element, {displayMode:false,throwOnError:false});$ = 90 seconds, the temperature at the wall with the radiative boundary condition is 215.713 K.

Table 14: Results for plane wall with radiative boundary condition.

Analytic	Bison	Error
215.713 K	216.217 K	0.234%

Conduction between Parallel Plates (Problem #11)

The problem is conceptually similar to Problem #7 but involves conductive heat transfer between the plates.

The distance between the plates is 1 in. The thermal conductivity, $var element = document.getElementById("moose-equation-0ceee85d-d9b8-439b-9252-b7ed16cc998c");katex.render("k", element, {displayMode:false,throwOnError:false});$ , of the gap is 0.1 W/in/K. This gives $var element = document.getElementById("moose-equation-48cb40af-ac89-45b7-8738-0dbe28d99e63");katex.render("h = k/L = 0.1", element, {displayMode:false,throwOnError:false});$ W/in $var element = document.getElementById("moose-equation-24aae958-aee6-41f1-a79a-b22dc067667d");katex.render("^2", element, {displayMode:false,throwOnError:false});$ /K.

The analytic solution for the flux is $var element = document.getElementById("moose-equation-1db22a0e-c18d-4f51-9d42-9db999887b24");katex.render(" q_1 = -h(T_1-T_2)", element, {displayMode:true,throwOnError:false});$

Table 15: Conduction across a Gap between Two Parallel Plates

Analytic	Bison	Error
-8525 W/m $var element = document.getElementById("moose-equation-e0c1cf74-44d3-4165-8d79-ea7eef09f8ac");katex.render("^2", element, {displayMode:false,throwOnError:false});$	-8524.98 W/m $var element = document.getElementById("moose-equation-6e980dcc-b6a4-4cd8-ba88-93c8d144b671");katex.render("^2", element, {displayMode:false,throwOnError:false});$	-0.00026%

Plane Wall with Heat Generation and Phase Change (Problem #12)

This problem is a wall with both surfaces insulated. The wall has changing heat capacity due to phase change. The left surface is at $var element = document.getElementById("moose-equation-3d05a803-1139-4324-b627-dd705d032cd6");katex.render("x=0", element, {displayMode:false,throwOnError:false});$ , and the right surface $var element = document.getElementById("moose-equation-6b7725a0-61f4-4043-ab73-1f996d3967d2");katex.render("x=L", element, {displayMode:false,throwOnError:false});$ . Also, the wall has internal heat generation $var element = document.getElementById("moose-equation-be8f08ad-ca2b-4f54-a2b0-4dc884971536");katex.render("Q", element, {displayMode:false,throwOnError:false});$ = 37.8 W/in $var element = document.getElementById("moose-equation-f3902516-b270-4d29-b460-81351eb65a6d");katex.render("^3", element, {displayMode:false,throwOnError:false});$ . The parameters for this problem are in Table 16.

Table 16: Parameters for plane wall problem with heat generation and phase change.

Symbol	Value
$var element = document.getElementById("moose-equation-7831e2b1-0139-499d-b0ab-a25651188c38");katex.render("k", element, {displayMode:false,throwOnError:false});$	100 W/in/K
$var element = document.getElementById("moose-equation-40f07c40-4dba-451f-ab73-5eab61ce6471");katex.render("c", element, {displayMode:false,throwOnError:false});$	4.2E5 J/kg/K
$var element = document.getElementById("moose-equation-c6f89415-d375-416b-801a-dc6acc5d42b3");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$	2.7E-5 kg/in $var element = document.getElementById("moose-equation-5aa355a7-cced-4650-b9ce-06843c56fe7f");katex.render("^3", element, {displayMode:false,throwOnError:false});$
$var element = document.getElementById("moose-equation-21faaecb-5d9c-4a95-8067-9d1957275798");katex.render("h_f", element, {displayMode:false,throwOnError:false});$	3.78E7 J/kg (latent heat)
$var element = document.getElementById("moose-equation-6e14f13f-db1f-4848-bd64-938720f14f08");katex.render("L", element, {displayMode:false,throwOnError:false});$	1.2 in
$var element = document.getElementById("moose-equation-6e944462-875e-4e12-b769-da0c2b916197");katex.render("T_i", element, {displayMode:false,throwOnError:false});$	60 $var element = document.getElementById("moose-equation-12c089b5-4879-4fea-83d8-4f0bcd3597c5");katex.render("\\degree", element, {displayMode:false,throwOnError:false});$ C
$var element = document.getElementById("moose-equation-a5c215fe-4798-4276-9bf5-5bd4499c2fc1");katex.render("T_S", element, {displayMode:false,throwOnError:false});$	400 K (solidus temperature)
$var element = document.getElementById("moose-equation-f24dee8e-1b65-4c76-9387-9ebb29612e32");katex.render("T_L", element, {displayMode:false,throwOnError:false});$	410 K (liquidus temperature)
$var element = document.getElementById("moose-equation-a2873ea4-811f-48f9-bbcd-b0929652c2b3");katex.render("c_f", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-0c88adaa-964f-48e2-b674-0bd819082e99");katex.render("h_f/(T_L-T_S)", element, {displayMode:false,throwOnError:false});$ = 3.78E6 J/kg/K (latent heat capacity)

The time to reach solidus temperature is $var element = document.getElementById("moose-equation-609226d6-6ee3-4208-9244-1e82c2614551");katex.render(" \\Delta t_S = (T_S-T_i)\\frac{\\rho c}{Q}", element, {displayMode:true,throwOnError:false});$

The time to reach liquidus temperature is $var element = document.getElementById("moose-equation-fc0db3f8-ac22-46cd-930a-512493f42083");katex.render(" \\Delta t_L = (T_L-T_S)\\frac{\\rho (c+c_f)}{Q}", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-204832cb-a851-46a6-a149-f36f9a39b348");katex.render("\\Delta t_S", element, {displayMode:false,throwOnError:false});$ = 30 s, and $var element = document.getElementById("moose-equation-db41ab98-f249-4b6d-8bb2-ab4bed1109dc");katex.render("\\Delta t_L", element, {displayMode:false,throwOnError:false});$ = 60 s.

The temperature at the wall at $var element = document.getElementById("moose-equation-2b29d9d4-e4af-41b6-bd63-acee9aa3dcda");katex.render("t", element, {displayMode:false,throwOnError:false});$ = 50 s is $var element = document.getElementById("moose-equation-55b5af50-b8cb-4a2f-9e4b-dc67fcb42b17");katex.render(" T = T_S+\\frac{Q}{\\rho(c+c_f)}(t-\\Delta t_S)", element, {displayMode:true,throwOnError:false});$

This gives $var element = document.getElementById("moose-equation-d311e27f-7c35-4ccb-a49c-7a316ecfe78a");katex.render("T", element, {displayMode:false,throwOnError:false});$ = 406.667 K.

Table 17: Results for plane wall with heat generation and phase change.

Analytic	Bison	Error
406.667 K	406.667 K	0%

References

F. P. Incropera and D. P. DeWitt. Fundamentals of Heat and Mass Transfer. Wiley, New York, fifth edition, 2001.

@book{INCROPERA2001,
    author = "Incropera, F. P. and DeWitt, D. P.",
    title = "Fundamentals of Heat and Mass Transfer",
    publisher = "Wiley",
    edition = "Fifth",
    year = "2001",
    address = "New York"
}

Overview
References