Summary: In this paper, we consider the following initial-boundary value problem \[ \begin{cases} u_t(x,t) = a(x)\Delta u(x,t) &\text{in}\;\Omega\times (0,T),\\ \frac{\partial_u(x,t)}{\partial \nu} = -b(x)g(u(x; t)) & \text{on}\;\partial \Omega \times (0,T),\\ u(x, 0) = u_0(x) &\text{in}\;\bar \Omega,\end{cases} \] where \(g :(0,\infty\to (0,\infty)\) is a \(C^1\) convex, nonincreasing function, \[ \lim_{s\to 0^+} g(s) =\infty, \quad \int_0 \frac{ds}{g(s)} < \infty, \] \(\Delta\) is the Laplacian, \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary \(\partial\Omega\), \(u_0 \in C^2(\bar \Omega)\), \(u_0(x)> 0\), \(x \in\bar \Omega\), \(a\in C^0(\bar \Omega)\), \(a(x) > 0\), \(x \in\bar \Omega\), \(b \in C^0(\partial\Omega)\), \(b(x) > 0\), \(x\in \partial\Omega\). Under some assumptions, we show that the solution of the above problem quenches in a finite time and estimate its quenching time. We also prove the continuity of the quenching time as a function of \(u_0\), \(b\) and \(a\). Finally, we give some numerical results to illustrate our analysis.