4. (14%) Solve the heat equation for 0 ＜x＜ L, t＞0, in which both ends are insulated: u_x(0,1)=0, u_x(L.t)=0 for t＞0；1C:u(x,0)=f(x)

1. (10%) Solve the initial value problem:

2. (10%) Solve the initial value problem: y"-2y'-8y=f(t)；y(0)=1.y'(0)=0.

3. (10%) Evaluate the line intcgral .dR with and C is parts of circle (x2+y2 =4, z=0) from (2,0,0) to (0.2.0).

5. (6%) Find the eigenvalues of matrix A: A =.

(a) (10%) In transport phenomena, we can divide into two parts and one part is related to the molecular transport processes fo corresponding cquations for momentum, heat and mass individually in one. for momentum, heat and mass. Please write down the dimensional case and give SI unit of correlated material's coefticient.

(b) (5%) The streum function y is a convenient pxrameter by which we can represent two- dimensional steady incompres ressible flow. Herein, we want to use stream function to describe the velocity in cylindrical low, vi and vo. Please desctibe v, eand yo by stream functiom,, in cylindrical coordinates] 

(a) (7%) Please give the definition of humidity (H), pereentage humidity (H,) and percentage relative humidity (Hr).

(b) (8%) A granular insoluble solid material wet with water is being dried in the cons stant-rate period in a pan 0.61 0.61 x 0.6I m2and the depth of material is 25.4 mm. The sides and bottoms are insulated. Air flows parallel to mi's and has a dry bulb temperature of 60℃ and wet bulb tempe rallel to the top drying surface at a velocity of 3.05 erature of 29.4℃. The pan contains 11.34 kg of dry solid having a free moisture content of 0.35 kg H2O/kg dry solid, and the material is to be dried in the constant-rate period to 0.22 kg H2O/kg dry solid. Please predict the drying time (unit: hours) needed.

8.(20%) Consider solvent A stored in a container and solvent A is Stream of B evaporating into gas B as the right diagram. The liquid level maintains at z=0 cm and the vapor pressure of solvent A in this level is 33 mmHg. A stream of gas am of gas B low ws slowly past the top of the container to maintain 755 mmHg at z=17.1 cm. The entire system is kept at 0℃ and 755 mmHg.Solvent A vapor and gas B are ass sumed to be ideal and the cross-scction area of the container is 0.82 cm2. If we can find that 0.0208 cm3of solvent A evaporated in 10 h after steady-statc. Please determi mine the value of diffusi of diffusion coefficien ? Where the density of solvent A is 1.59 g/cm3, the molecular weight is 154 g/gmol and gas constant is 82.05 (cm3●atmvgmol●K). Please state clearly the assumption when you solve the problem. 

(b) (6%), (7%) Please give their (Equations (2) and (3)) matching engineering (or physical) examples with discussion, respectively.

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