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I am trying to solve a set of partial differential equations but the system returns the following error -:

Error using pdepe (line 293)

Spatial discretization has failed. Discretization supports only parabolic and elliptic equations, with flux term involving spatial derivative.

The code with the local functions is as follows -

clc;

clear all;

global k_10 k_20 E2_by_R E1_by_R h_1 h_2 v_r rho_r C_pr U_1 d_1 d_2 U_2 n d_3 rho_c C_pc v_c L rho_w C_pw ;

%% parameters

d_1 = 0.02; %m

d_2 = 0.024; %m

d_3 = 1; %m

n = 1200;

L = 6; %m

rho_r = 985; %kg/m3

rho_w = 7800; %kg/m3

rho_c = 998; %kg/m3

C_pr = 4.05; %kJ/kg*K

C_pw = 0.71; %kJ/kg*K

C_pc = 4.18; %kJ/kg*K

U_1 = 2.8; %kJ/m2*K*s

U_2 = 2.56; %kJ/m2*K*s

k_10 = 5.61e16; %/s

k_20 = 1.128e16; %/s

E1_by_R = 13477; %K

E2_by_R = 15290; %K

h_1 = 5.8e4; %kJ/kmol

h_2 = 1.8e4; %kJ/kmol

c_A0 = 2.85; %kmol/m3

T_r0 = 323; %K

T_c0 = 293; %K

q_r = 0.05;

q_c = 0.35;

f_r = n*pi*d_1*d_1/4;

f_c = pi*(d_3*d_3 - n*d_2*d_2)/4;

v_r = q_r/f_r; %m/s

v_c = q_c/f_c;

z = linspace(0,6,100);

t = linspace(0,1000,1000);

m = 0;

sol = pdepe(m,@pdefun,@pdeic,@pdebc,z,t);

y1 = sol(:,:,1);

y2 = sol(:,:,2);

y3 = sol(:,:,3);

y4 = sol(:,:,4);

y5 = sol(:,:,5);

function [c,f,s] = pdefun(z,t,y,dydz)

global k_10 k_20 E2_by_R E1_by_R h_1 h_2 v_r rho_r C_pr U_1 d_1 d_2 U_2 n d_3 rho_c C_pc v_c L rho_w C_pw;

c_A = y(1)

c_B = y(2)

T_r = y(3)

T_w = y(4)

T_c = y(5)

k_1 = k_10*exp(-E1_by_R/T_r);

k_2 = k_20*exp(-E2_by_R/T_r);

h_r = h_1*k_1*c_A+h_2*k_2*c_B;

s1 = -k_1*c_A;

s2 = (k_1*c_A-k_2*c_B);

s3 = (h_r/(rho_r*C_pr) - 4*U_1*(T_r-T_w)/(d_1*rho_r*C_pr));

s4 = 4*(d_1*U_1*(T_r-T_w)+d_2*U_2*(T_c-T_w))/(rho_w*C_pw*(d_2*d_2-d_1*d_1));

s5 = ((4*n*d_2*U_2*(T_w-T_c))/((d_3^2 - n*d_2^2)*rho_c*C_pc));

c = [1;1;1;1;1]

f = [-v_r*c_A;-v_r*c_B;-v_r*T_r;0;v_c*T_c]

s = [s1;s2;s3;s4;s5]

end

function u0 = pdeic(z)

u0 = [2.85;0;323;307.30;293];

end

function [pl,ql,pr,qr] = pdebc(zl,yl,zr,yr,t)

pl = [yl(1)-2.85;yl(2)-0;yl(3)-323;0;0];

pr = [0;0;0;0;yr(5)-293];

ql = [0;0;0;1;0];

qr = [0;0;0;1;0];

end

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