Chemical Engineering 6453 Heat Transfer Prof. Geoff Silcox Spring 2005 Solution to Homework Assignment 7 Due Wednesday, 2 March, by 17:00.

Problem 1 For a duct that is heated on both the top and bottom with qbot = qtop, calculate the Nusselt number for fully developed, constant property, laminar flow. The width of the duct is much greater than its height. Use the numerical approach discussed in class. You may start with the MATLAB file, one_dq.m, that is posted on the class web site. That file models a specified flux on the bottom and a well insulated top. Solution If the duct is heat from above and below with uniform flux (W/m2) qw, the source term (W/m3) for control volume i becomes

2 w ii

q wSD w

= − (1)

Because the problem is symmetrical, the temperatures at the top and bottom walls are equal. Call that temperature Tb. The energy balance for control volume n becomes

1 18 9 03

b n n n nn

T T T T Tk k S yy y

− −− + −+ + ∆ =

∆ ∆ (2)

Our approach is to take qw = 1 and to set Tb = 0. This prevents the occurrence of a singular matrix (det(A) = 0) and is equivalent to the choice of a particular axial location in the duct. The calculated Nusselt numbers for heating from one side and both sides are summarized in Table 1 and are compared to values from Table 4.5, p. 307 of your text. All numerical calculations were performed with 20 control volumes.

Table 1 Comparison of published and numerically calculated Nusselt numbers for a one-dimensional duct heated with uniform axial flux from one side and two sides. In the former, the unheated surface is insulated.

one

side top and bottom

text 5.385 8.235 calculated 5.386 8.242

The calculated dimensionless temperature profile is shown in Figure 1. The profile is symmetrical as required by the boundary conditions. The dimensionless temperature is

( )( ) b

b m

T T yyT T

θ−

=−

(3)

The MATLAB code that produced Figure 1 is one_dqtb and a link is provided.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4Dimensionless temperature profile

Distance from bottom, m

Tem

pera

ture

Figure 1 Dimensionless temperature profile for one-dimensional duct with uniform axial flux on the top and bottom.

Problem 2 For a duct that is heated on both the top and bottom with Tbot = Ttop, calculate the Nusselt number for fully developed, constant property, laminar flow. The width of the duct is much greater than its height. Use the numerical approach discussed in class. You may start with the MATLAB file, one_dt.m, that is posted on the class web site. That file models a specified temperature on the bottom and a well insulated top. Solution If the duct is heat from above and below with specified temperature Tb, the source term (W/m3) for control volume i becomes

2b w ii

b m

T T q wST T D w⎛ ⎞−

= −⎜ ⎟−⎝ ⎠ (4)

The energy balance for control volume n becomes

1 18 9 03

b n n n nn

T T T T Tk k S yy y

− −− + −+ + ∆ =

∆ ∆ (5)

Our approach is to set Tb = 0, guess all interior temperatures, and calculate qw. This fixes the value of qw and is equivalent to the choice of a particular axial location in the duct. Because of (4) the problem is nonlinear and we must iterate to obtain a converged solution (see one_dttb). The calculated Nusselt numbers for heating from one side and both sides are summarized in Table 2 and are compared to values from Table 4.5, p. 307 of your text. All numerical calculations were performed with 20 control volumes.

Table 2 Comparison of published and numerically calculated Nusselt numbers for a one-dimensional duct heated with uniform axial temperatures from one side and two sides. In the former, the unheated surface is insulated.

one

side top and bottom

text 4.861 7.541 calculated 4.859 7.534

The calculated dimensionless temperature profile is shown in Figure 2. The profile is symmetrical as required by the boundary conditions. The MATLAB code that produced Figure 2 is one_dttb and a link is provided.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4Dimensionless temperature profile

Distance from bottom, m

Tem

pera

ture

Figure 2 Dimensionless temperature profile for one-dimensional duct with uniform and equal temperatures on the top and bottom.