Question

the curing of concrete is a chemical reaction involving the hydration of Portland cement. at any time, the rate of increase of the compressive strength S of the concrete is proportional to the difference between its final strength (in this case 5000psi) and the current value of S at any time. a. Write a differential equation for dS/dt , with k unknown at this point. ( the value of k will not be determined until part c) b. Solve the differential equation to find a function describing the strength at any time S(t) assuming an intial condition of zero strength at t=0 c. After 28days, if cured correctly, the strength of the concrete is typically 95% of its final strength. Use S(28)=4750 to evaluate k and complete the model for S(t)

Answer 1

Let final compressive strength = S'

Then, if dS(t)/dt is proportional to the difference between the final strength and the current strength

dS(t)/dt = K(S'final - S(t))

(K = constant of proportionality)

Rearrange the above equation, we get

dS/(S - S'final) = -Kdt

Perform integration on both sides,

ln(S - S'final) - ln(So - S'final) = -Kt

(So = strength at zero time)

We can also write the above equation as follow

ln(S - S'final / S0 - S'final) = -Kt

S(t) = S'final + (So - S'final)exp(-Kt)

We know S(0) = So and S'final = 5000psi

Therefore, S(t) = S'final - S'finalexp(-Kt)

                         = (5000psi)(1-exp(Kt))

Now,

S(28days) = 4750psi = (5000psi)(1 - exp(K x 28days))

0.95 = 1 - exp(-K x 28days)

0.05 = 1/20 = exp(-K x 28days)

ln(1/20) = -Kx28

K = 0.107/day

Therefore,

S(t) = (5000psi)(1 - exp(-0.107t/day))