Introduction to Heat and Mass Transfer. Week 9

Size: px

Start display at page:

Download "Introduction to Heat and Mass Transfer. Week 9"

Erick Tracy Ford
5 years ago
Views:

1 Introduction to Heat and Mass Transfer Week 9

2 補充! Multidimensional Effects Transient problems with heat transfer in two or three dimensions can be considered using the solutions obtained for one dimensional transient cases Usually multiple dimensions can be considered as product of the following one-dimensional solutions: S x, t T x, t T T T i semi-infinite solid P x, t T x, t T i T T plane wall C r, t T r, t T i T T infinite cylinder

3 Multidimensional Effects (contd.)

5 Multidimensional Effects (contd.)

6 HW # 5 prob. 7 A stainless steel cylinder (k=16.3 W/mC, α=0.44x10-5 m 2 /s, ρ =7817kg/m 3, c= 460 J/kgC) is heated to a uniform temperature of 220C and then allowed to cool in an environment where the air temperature is maintained constant at 20C. The convection heat transfer coefficient may be taken as 180 W/m 2 C. The cylinder has a diameter of 12 cm and a length of 18 cm. Calculate the temperature of the geometric center of the cylinder after a time of 10 min using product solution method and Heisler-Grober charts. Also calculate the heat loss.

7 HW # 5 will due on 11/15 (Thursday), right before the class! Late HW will not be accepted!!

8 Transient Analysis Strategy For a general transient conduction problem, we can follow the following procedure:» Compute Biot number and check whether Bi < 0.1» If Bi < 0.1, then we can usually implement lumped system analysis with reasonable accuracy» Otherwise we compute required constants as function of Biot number and implement approximate 1D analytical solutions for large plane walls (small thickness) or infinitely long cylinders (l/r o >10) or spheres (We can implement Heisler-Grober charts for the these standard shapes)» If multidimensional effects are present, then we consider the given solid as product of relevant 1D analytical solutions for standard shapes

9 Questions What is the physical significance of Biot number? What is the physical significance of Fourier number? What is the product solution method? How can it be used for multidimensional systems?

10 Closure Coverage thus far..» talked about non-dimensionalization of heat diffusion equation to solve transient conduction problems» discussed analytical solutions for large plane wall, infinite cylinder and sphere as well as semi-infinite solid» considered multidimensional heat transfer treatment

11 Closure (contd.) Analytical solutions for one dimensional transient heat conduction through large plane wall, infinite cylinder and sphere using non-dimensional variables Use of Heisler-Grober charts for solving the above heat conduction problems f Similarity solutions for semi-infinite solids Multidimensional heat transfer analysis using the method of product solutions x, t, Bi

12 Next Topic Transient Conduction» Numerical Method Finite Difference Formulation Explicit Method Stability Criterion Implicit Method

13 Numerical Method For transient heat transfer problems:» Lumped capacitance analysis restricted to few cases where temperature gradients absent» Analytical solutions applicable to few simple geometries under benign boundary conditions Numerical methods useful for solving more complicated unsteady problems For two dimensional, transient conditions with constant properties and no thermal energy generation, we solve: T T T t x y 2 2

14 Dy Finite Difference Formulation i, j+1 - If Dx = Dy Uniform Mesh i-1, j i, j i+1, j Spatial temperature derivatives just like 2D steady conduction Dx i, j-1 How do we approximate time derivative of temperature? Explicit Method Implicit Method Two Dimensional Mesh

15 Explicit Method Using forward difference approximation for the temporal temperature derivative, we can write: T t i, j 1 T 2 For explicit method, we use temperature values at previous time step when using central difference approximations for the spatial temperature derivatives 2 T P1 T P i, j i, j Dt 1,, 1, x Dx y Dy 2 i, j explicit T T T P P P 2 i j i j i j 2 2 T 2 i, j explicit T 2T T P P P i, j1 i, j i, j1 2

16 Explicit Method (contd.) Substitute into heat diffusion equation to obtain discretized equation for any interior node: explicit P1 P P P P P T Fo T Fo T T T T 1 4 i, j i, j i1, j i1, j i, j1 i, j1 In the above equation, RHS depends only on temperature at previous time step Separate discretized equations at boundary nodes No solver required because current temperature depends only on previous temperatures Explicit method very easy to implement numerically

17 Stability Criterion Although the explicit method is rather simple it is almost never used in practice since it is conditionally stable Using general stability analysis, we can show that the time step is restricted to obtain bounded solutions von Neumann stability limit for 2D: Fo explicit 14Fo explicit Dt As Dx refined to reduce error, Dt becomes increasingly explicit restrictive and computationally cumbersome Dx 4 2

18 Implicit Method Using forward difference approximation for the temporal temperature derivative, we can write: For implicit method, we use temperature values at current time step when using central difference approximations for the spatial temperature derivatives 2 T P1 P1 P1 1,, 1, x Dx y Dy 2 i, j implicit T t T 2T T i j i j i j 2 i, j 1 2 T P1 i, j i, j 2 T Dt T 2 i, j P implicit T 2T T P1 P1 P1 i, j1 i, j i, j1 2

19 Implicit Method (contd.) Substitute into heat diffusion equation to obtain discretized equation for any interior node: implicit P P 1 P 1 P 1 P 1 P 1 T Fo T Fo T T T T 1 4 i, j i, j i1, j i1, j i, j1 i, j1 In the above equation, RHS depends on temperatures at current time step Separate discretized equations at boundary nodes The resulting linear system of simultaneous equations solved using either direct (e.g. Matrix Inversion) or iterative (e.g. Gauss Seidel) techniques Implicit method rather difficult to implement numerically

20 Implicit Method (contd.) Although implicit method is difficult to code, it is unconditionally stable i.e. large time steps are allowed However, Dt should be chosen such that the underlying physics of the given problem is resolved correctly Numerical methods (explicit or implicit) solve discretized algebraic equations that are approximations of the original PDE the accuracy increases using mesh refinement (smaller Dx) and finer time steps (smaller Dt)

21 Example Determine its stability criterion. (using Finite Volume Method)

22 Questions Is there any limitation on the size of the time step in transient heat conduction problems using (a) explicit method and (b) implicit method? Consider transient one dimensional heat conduction through a plane wall where both sides are at specified temperature. For explicit method, express the stability criterion.

24 HW # 6 prob. 1 Determine its stability criterion. (using Finite Volume Method)

25 Closure Coverage thus far..» finite difference formulation for transient heat conduction problems in two dimensions» numerical methods for solving transient heat conduction problems by space-time discretization Explicit Method Implicit Method» von Neumann stability criterion for explicit method

26 Closure (contd.) Finite difference formulation of heat conduction problems using central difference approximation or Finite volume formulation of heat conduction problems using conservation of energy and discretized equations for interior nodes Finite volume formulation of heat conduction problems using conservation of energy and discretized equations for boundary nodes Use of direct and iterative techniques for solving the resulting set of algebraic equations

27 Next Topic Convective Heat Transfer» Convection Mechanism Heat Transfer Mass Transfer» Convection Boundary Layers Velocity Boundary Layer Thermal Boundary Layer Concentration Boundary Layer

28 Convection Mechanism At the solid-fluid boundary, any transport occurs only due to diffusion process (random motion) With increase in fluid velocity away from the boundary, bulk motion increases (ordered motion) Recall: Convection is the cumulative result of diffusion and bulk motion Convection depends on fluid properties, characteristics of solid surface and type of fluid flow

29 Convection Mechanism (contd.) Although convection heat transfer is complicated, the governing rate equation is rather simple q h T T conv conv s " Considering an irregular shaped body, we can write: q h A T T conv conv s s h 1 S A conv conv s A s h da In general, convective heat transfer coefficient varies along the body

30 Convection Heat Transfer Coefficient h It s a constant which depends on conditions of surface geometry, the nature of the fluid motion, and the fluid properties. Process h (W/m 2 K) Free Convection Gases 2-25 Liquids Forced Convection Gases Liquids h free convection < h forced convection ; h gases < h liquids

31 Convection Heat Transfer Coefficient h

32 Example A circular hot gas jet at T is directed normal to a circular plate of radius r o maintained at uniform temperature T s. The gas flow over the plate is axi-symmetric and the local convection coefficient is given as h(r) = a + br n, where a, b and n are known constants.» Determine the convective heat transfer rate to the plate in terms of T, T s, r o, a, b and n

33 Convection Mechanism (contd.) Convection mass transfer is very similar to heat transfer; the governing rate equation where, N h C C A m A s A ",,» N A = Molar flux of species A at the surface (kmol/m 2 -s)» h m = Convective mass transfer coefficient (m/s)» C A,s and C A, = Molar concentrations of species A at the surface and in the free stream (kmol/m 3 ) Considering an irregular shaped body, we can write: N h A C C A m s A s A,, h 1 S A m m s A s h da

34 Example

35 Convection Boundary Layers Computation of convective heat transfer coefficient requires knowledge of temperature field To obtain temperature field, we need to solve the energy equation through the boundary layer To solve energy equation, we need to calculate the flow field i.e. solve momentum and continuity equation through the boundary layer Concepts related to boundary layer development and growth are necessary for convective heat transfer problems

36 Velocity Boundary Layer No slip condition requires fluid adjacent to the surface to have zero velocity Owing to viscosity and shear stresses, the fluid layers away from the wall are retarded Through the velocity boundary layer, velocity gradients and shear stresses are large Outside the boundary layer, velocity gradients and shear stresses are relatively small Velocity boundary layer grows with distance from the leading edge

37 Velocity Boundary Layer (contd.) y u Leading Edge x u = 0; x = 0 y u y when 0.99 u

38 Velocity Boundary Layer (contd.) For a Newtonian fluid, we can write: s u y Local friction coefficient along the surface is given as: C f Friction coefficient extremely important in determination of drag forces and losses s 2 u y0 2

39 Thermal Boundary Layer No temperature jump condition requires fluid adjacent to the surface to have same temperature as the surface Owing to temperature gradients depending on fluid motion, the fluid layers away from the wall have lower temperature Through the thermal boundary layer, temperature gradients and diffusive transfer rates are large Outside the boundary layer, temperature gradients and diffusive transfer rates are relatively small Thermal boundary layer grows with distance from the leading edge

40 Thermal Boundary Layer (contd.) T y t Leading Edge x when 0.99 s t y y T = T s ; x = 0 T T s T T

41 Thermal Boundary Layer (contd.) Using Fourier s law at the surface, we can write: The above heat transports away from the surface via advection process Local convective heat transfer coefficient along the surface is given as: h q " s k fluid T y y 0 T k " fluid q y s T T T T s s y0

42 Question What is a Newtonian fluid? What fluid property is responsible for the development of velocity boundary layer? What fluid property is responsible for the development of thermal boundary layer?

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