Introduction to Heat and Mass Transfer. Week 8

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1 Introduction to Heat and Mass Transfer Week 8

2 Next Topic Transient Conduction» Analytical Method Plane Wall Radial Systems Semi-infinite Solid Multidimensional Effects

3 Analytical Method Lumped system analysis valid only when temperature gradients within the solid ignored In general, we must solve heat diffusion equation with appropriate spatial coordinates along with the transient term For one-dimensional heat conduction with constant properties and no thermal energy generation 2 T 1 T 2 x t Need two boundary conditions and one initial condition

4 Analytical Method (contd.) T(x,0) = T i T(r,0) = T i h, T h, T h, T h, T Sphere 2L x x * = x/l T(r,0) = T i 2r o r r * = r/r o Large Plane Wall h, T Long Cylinder r o r * = r/r o

5 Analytical Method (contd.) The solution consists of spatial and temporal variables as well as thermal properties and other parameters Non-dimensional analysis helps in a more simplified solutions via casting relevant variables into groups We will consider the following non-dimensional groups:» Temperature» Spatial Coordinate» Temporal Coordinate T T T T i x i x L t t 2 L r r r o

6 Analytical Method (contd.) With the above non-dimensional groups, we need to solve: 2 2 x t Boundary conditions can also be non-dimensionalized using the same groups In each case, we obtain solutions that depend on constants which are functions of Biot number These constants are listed in Table 5.1 for the cases of large plane wall, infinite (long) cylinder and sphere

7 Plane Wall Exact solution can be written as: C n plane wall plane wall C Fo x n 4sin 2 exp cos 1 n n n n For Fo > 0.2, the above infinite series converges and an approximate solution can be given as: 2 sin 2 plane wall n n tan Bi n n C 2 exp Fo cos x planewall plane wall cos x 0 1 plane wall C exp Fo

8 Plane Wall (contd.) Total energy transfer from the plane wall at any time t can be written as: where,» Q 0 = Maximum possible energy transfer Maximum possible energy transfer occurs at infinitely long time Q planewall Q sin Q mc T T 0 p i

9 Radial Systems Exact solutions for infinite (long) cylinder and sphere available just like plane wall For Fo > 0.2, the infinite series solutions converge and an approximate solutions can be given as: exp cylinder C 2 Fo J r cylinder cylinder exp J r sphere C C sin r 2 1 exp Fo 1 1 r 1 2 Fo C exp Fo sphere 0 sin r r 1 1 sphere

10 Radial Systems (contd.) Total energy transfer from the infinite cylinder or sphere at any time t can be written as: where, Q sphere Q Q 1 2 J cylinder 0 Q sin cos » Q 0 = Maximum possible energy transfer Maximum possible energy transfer occurs at infinitely long time

12 補充! Analytical Method (contd.) Use of Heisler-Grober charts (plane wall)

13 Analytical Method (contd.) Use of Heisler-Grober charts (plane wall)

14 Analytical Method (contd.) Use of Heisler-Grober charts (plane wall)

15 Example Steel is annealed to relieve stresses to make it less brittle. Consider a 100-mm thick plate that is initially at an uniform temperature of 300C and is heated (on both sides) in a gas-fired furnace for which the surrounding temperature is 700C and heat transfer coefficient is 500 W/m 2 -K.» How long does it take to attain a minimum temperature of 550C in the plate? Use of one-term approximation equations!

16 Example A spherical hailstone 5 mm in diameter is formed in a cloud at -30C. The stone begins to fall through warmer air at 5C. Convection heat transfer coefficient can be considered to be 250 W/m 2 -K and hailstone can be modeled as ice.» How long does it take before the outer surface of the hailstone begins to melt?» What is the temperature of hailstone s center at this instant?» How much energy (J) is transferred to the hailstone? Use of Heisler-Grober charts!

17 HW # 5 prob. 4 Steel is annealed to relieve stresses to make it less brittle. Consider a 100-mm thick plate that is initially at a uniform temperature of 300C and is heated (on both sides) in a gas-fired furnace for which the surrounding temperature is 700C and heat transfer coefficient is 500 W/m 2 -K.» How long does it take to attain a minimum temperature of 550C in the plate? Use of Heisler-Grober charts!

18 HW # 5 prob. 5 A spherical hailstone 5 mm in diameter is formed in a cloud at -30C. The stone begins to fall through warmer air at 5C. Convection heat transfer coefficient can be considered to be 250 W/m 2 -K and hailstone can be modeled as ice.» How long does it take before the outer surface of the hailstone begins to melt?» What is the temperature of hailstone s center at this instant?» How much energy (J) is transferred to the hailstone? Use of one-term approximation equations!

19 Semi-infinite Solid A solid which extends infinitely in all directions except one particular direction Thermal changes at the surface do not transmit quickly through the remaining body Using a similarity variable (x/t 1/2 ), the PDE governing heat transfer through the semi-infinite solid reduced to an ODE expressed in terms of the similarity variable Note that the similarity variable dictates the following peculiar behavior for semi-infinite solids:» As x, we can consider t 0» As t, we can consider x 0

20 Semi-infinite Solid (contd.)

21 Semi-infinite Solid (contd.) Solutions usually obtained in terms of error function and complementary error function 2 x exp 2 erf x u du 0 erfc x 1erf x Error functions tabulated in Table B-2 Three common surface conditions of practical interest:» Constant surface temperature» Constant surface heat flux» Constant surface convection

22 Semi-infinite Solid (contd.)

23 Semi-infinite Solid (contd.) Constant Surface Temperature Constant Surface Heat Flux T x, t T x s erf T T 2 t Constant Surface Convection i s 1/ 2 2q " t x 2 qx " 0 x 0 T x, t T exp erfc i k 4 t k 2 t 2, T x t Ti x hx h t x h t erfc exp erfc 2 T T 2 t k k 2 t k i

24 Semi-infinite Solid (contd.)

25 HW # 5 prob. 6 A thick oak wall initially at 25C is suddenly exposed to surrounding of combustion products where the temperature is 800C and heat transfer coefficient is 20 W/m 2 -K.» Determine the time of exposure required for the surface to reach the ignition temperature of 400C Use of Fig. 5.8!

Introduction to Heat and Mass Transfer. Week 8

Introduction to Heat and Mass Transfer Week 8 Next Topic Transient Conduction» Analytical Method Plane Wall Radial Systems Semi-infinite Solid Multidimensional Effects Analytical Method Lumped system analysis