In determining the temperature distribution in a cylinder of radius R (see Unste

Question

In determining the temperature distribution in a cylinder of radius R (see Unsteady Heat Transfer II in Section 10.2.3), the following integral involving the Bessel function, J_0, arises: I_m = integral_0^R r J_0 (lambda_m r) dr where 1ambda_m is determined by the equation J_0 (lambda R)/J_1 (lambda R) - lambda R k/hR = 0 where h is the convective heat transfer coefficient and k is the thermal conductivity of the cylinder material. In Project 10.3 the values in Table P5.2 of lambda_m were obtained Use MATLAB's quad function to determine I_m. Take R = 0.12 m. Construct a table of I_m vs. index m.

Explanation / Answer

We can have Bessel function using the Matlab provided bessel function.

Given lambda = 14.9676, 37.2605, 37.2605, 61.8018, 87.1761

For each value of lambda we need to integrate.

CODE:
>> lambda = 14.9676;
>> F = @(r)r.*besselj(14.9676,r);
>> quad(F,0,0.12)
ans =

3.6586e-034
  
Change the value of lambda

>> lambda = 37.2605;
>> F = @(r)r.*besselj(lambda,r);
>> quad(F,0,0.12)

ans =

3.3648e-093
  
>> lambda = 61.8018;
>> F = @(r)r.*besselj(lambda,r);
>> quad(F,0,0.12)
ans =

7.1119e-165
  
>> lambda = 87.1761;
>> F = @(r)r.*besselj(lambda,r);
>> quad(F,0,0.12)
ans =

2.0194e-243

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