353 WK9TPCSWEB.TEX
ME M.M. Yovanovich
Week 9
Makeup lecture 3. Week 8 lecture summary and several pages of summary of convective heat transfer correlations are available in Engineering Photocopy Center. Pick up and bring to future lectures. See Appendices for special functions. Table B.2 on page 857 for erf w and 0 w 3. Table B.5 on page 860 for modi ed Bessel functions: I0x; I1x; K0x; K1 x for 0 x 10. The rst ve roots of characteristic equation for plane wall n sin n = Bi cos n for a range of values of the Biot number. See Maple worksheets on ME 353 Web site how to calculate Bessel functions, and to obtain the roots of the characteristic equations for plane wall, long circular cylinder and solid sphere. Multidimensional systems: i nite circular cylinder with side cooling and end cooling, ii rectangular plate, and iii cuboid parallelopiped. See Section 5.8. Finite circular cylinder of diameter D and length 2L. Temperature excess depends on r; z; t. There are two sets of Biot and Fourier numbers: Bi1 = h1 L=k; F o1 = t=L2 and Bi2 = h2 D=2=k; F o1 = t=D=22 . Here the subscripts 1; 2 denote cooling at two ends and side respectively. Dimensionless temperature excess is obtained from the plane wall and long circular cylinder solutions: ! ! ! =
i cp i c i p
Lecture 2
Lecture 1
and heat loss fraction is obtained from ! ! ! ! ! Q Q Q Q Q = Q + Q , Q Q Qi cp i c i p i c i p and Qi = cpD=22 2L i.
Rectangular plate of dimensions: 2L1 ; 2L2; 2L3 corresponding to the x; y; zcoordinates respectively. The dimension 2L3 2L1 and 2L2 . Heat transfer coe cients are h1 on the L2L3-surfaces and h2 on the L1L3-surfaces. The surfaces perpendicular to the z,coordinate are adiabatic. The two sets of Biot and Fourier numbers are: Bi1 = h1L1=k; F o1 = t=L2 and Bi2 = h2L2=k; F o2 = 1 t=L2 . 2 Dimensionless temperature excess which depends on x; y; t is obtained from the plane wall solution applied twice: ! ! ! =
i p1 p2 i p1 i p2
and heat loss fraction is obtained from ! ! ! ! ! Q Q Q Q Q = Q + Q , Q Q Qi p1 p2 i p1 i p2 i p1 i p2 and Qi = cp2L12L22L3 i. See example 5.6 transient conduction in nite cylinder and Maple worksheets on ME 353 Web site. Cuboid of dimensions: 2L1 ; 2L2; 2L3 corresponding to the x; y; z-coordinates respectively. Heat transfer coe cients are h1 on the L2L3-surfaces, h2 on the L1 L3 -surfaces, and h3 on the L1 L2 -surfaces. The three sets of Biot and Fourier numbers are: Bi1 = h1L1=k; F o1 = t=L2 , Bi2 = h2L2=k; F o2 = t=L2 and 1 2 Bi3 = h3 L3 =k; F o3 = t=L2 . 3 Dimensionless excess temperature which depends on x; y; z; t is obtained from the plane wall solution applied three times: ! ! ! ! =
i p1 p2 p3 i p1 i p2 i p3
and heat loss fraction is obtained from ! ! 2 ! 3 ! 2 ! 32 ! 3 ! Q Q Q Q Q Q Q = Q + Q 41 , Q 5+ Q 41 , Q 5 41 , Q 5 Qi p1 p2 p3 i p1 i p2 i p1 i p3 i p1 i p2 and Qi = cp2L12L22L3 i. Approximations of Bessel functions J0x; J1x based on trapezoidal rule applied to integral forms of the Bessel functions. Also see B.4, page 859 for 2
short tables of J0x; J1x for 0 x 2:4. The following relations are based on 4 panels: ! 1 + 1 cos p + 1 cosx x J0x = 4 2 4 2 and ! ! 1 sin p + + 1 sinx , 1 cos p + x x J1x = 4 4 4 2 4 2 4 These approximations can be used in the single term approximate solutions for long cylinders.
Lecture 3
Convective Heat Transfer. The four chapters which deal with this topic are: Chapter 6: Review of uid mechanics and de nitions. Chapter 7: External forced convection. Chapter 7: Internal forced convection. Chapter 8: External and internal natural convection. Convective heat transfer analysis is complex because, in general, one must solve 6 equations simultaneously to nd the three velocity components: u; v; w, the temperature T and the pressure: P . The six equations are: Continuity equation 1 Momentum equations 3 Energy equation 1 Equation of state 1 Correlation equations are based on i approximate analytical solutions, ii experiments and iii numerical solutions. The correlation equations relate the Nusselt number to other dime...