Chemical Engineering 6453 Heat Transfer Prof. Geoff Silcox Spring 2005 Homework Assignment 4 Due Friday, 11 February, by 17:00.

Problem 1 The motion of motile bacteria can be characterized by an effective diffusivity, D. A convenient way of measuring the diffusivity is to place one end of a capillary tube into a suspension of the bacteria, wait a few minutes, and count the number of bacteria in the tube. Before insertion into the solution, the tube is filled with the same liquid culture medium in which the bacteria are swimming. The number of bacteria in the tube can be compared with theory and the effective diffusivity can be calculated. Two sets of data have been obtained for a 1-µL tube that is 32 mm long.

Set One Concentration in bulk suspension: C0 = 7 x 107 bacteria/cm3 Time interval: T = 2, 5, 10, 12.5, 15, 20 min. Number of bacteria in tube: N = 1800, 3700, 4800, 5500, 6700, 8000

Set Two C0 = 2.5, 4.6, 5.0, and 12.0 x 107 bacteria/cm3 T = 10 min. N = 1350, 2300, 3400, 6200

a) Derive a theoretical expression that relates D, N, C0, T, and the cross-sectional area of the tube, A. b) Plot each set of data and fit the theoretical expression to the data. Report D for each set of data. c) Using the values of D obtained in part (b), justify the use of the theoretical expression derived in part (a).

Problem 2 Use Duhamel's superposition integral (see Lesson 10) to obtain T(t) for

0

0

( )

(0)( ) cos( )

dTc hA T TdtT T

T t T t

ρ

θ ω

∞

∞ ∞

Ω = − −

=

= +

Comment on the ability of a thermometer to track the changing temperature. How small does the thermometer have to be to provide accurate temperatures?