Question 4 of 8: Scale-Up of Regenerator Stoves — Dimensional Analysis

Question 4: Scale-Up of Regenerator Stoves — Dimensional Analysis (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Dependent variable $\Delta\theta_{g,t}$ (temperature rise) and 9 independent variables $\{\Delta\theta_0,\,t,\,C_g,\,C_s,\,U_0,\,k,\,\mu_0,\,\rho,\,L,\,d\}$, spanning 4 primary dimensions $M,L,T,\Theta$.

Find. The dimensionless functional form for $\Delta\theta_{g,t}/\Delta\theta_0$; whether the variable list is complete for ceramic spheres; and the ratio of transit times $t_2/t_1$ for the doubled-scale regenerator under the stated flow condition.

Approach. Form the natural temperature-ratio group directly ($\Delta\theta_{g,t}/\Delta\theta_0$, both variables sharing dimension $\Theta$), then apply Buckingham's $\Pi$-theorem to the remaining 9 variables using $\rho$, $U_0$, $d$, $C_g$ as repeating variables to span $M,L,T,\Theta$.

1. Dimensions of each variable. $t:T$; $C_g,C_s: ML^{-1}T^{-2}\Theta^{-1}$ (energy per volume per K); $U_0:LT^{-1}$; $k:MLT^{-3}\Theta^{-1}$; $\mu_0:ML^{-1}T^{-1}$; $\rho:ML^{-3}$; $L,d:L$.
2. Variable/dimension count. 9 independent variables, 4 dimensions $\Rightarrow$ Buckingham predicts $9-4=5$ independent $\Pi$ groups (beyond the already-formed $\Delta\theta_{g,t}/\Delta\theta_0$).
3. Form the groups (repeating variables $\rho,U_0,d,C_g$). Combining each remaining variable with the repeating set to cancel $M,L,T,\Theta$ gives the classical named groups: $$\Pi_1=\dfrac{\rho U_0 d}{\mu_0}=Re_d,\qquad \Pi_2=\dfrac{\mu_0 C_g}{\rho k}=Pr,\qquad \Pi_3=\dfrac{C_s}{C_g},\qquad \Pi_4=\dfrac{L}{d},\qquad \Pi_5=\dfrac{U_0 t}{L}$$ (each verified dimensionless by direct substitution of Step 1's dimensions.)
4. Functional form. $$\boxed{\dfrac{\Delta\theta_{g,t}}{\Delta\theta_0}=\Phi\!\left(Re_d,\ Pr,\ \dfrac{C_s}{C_g},\ \dfrac{L}{d},\ \dfrac{U_0 t}{L}\right)}$$
5. Sufficiency for ceramic spheres. The list covers convective/conductive transport (via $Re_d,Pr,k$) and thermal-mass matching (via $C_s/C_g$), but contains no radiative variables (emissivity $\varepsilon$, Stefan–Boltzmann $\sigma$). At the high gas/sphere temperatures typical of regenerative stoves, radiation is not always negligible, and ceramic (refractory) spheres commonly have different emissivity — and can be partially transparent — compared with metal spheres. The list is therefore not fully sufficient for ceramic spheres; a radiation-related group (e.g. $\varepsilon\sigma\theta^3 d/k$) would need to be added if radiative exchange is significant.
6. Geometric similarity, doubled scale. Full geometric similarity: $L_2=2L_1$, $d_2=2d_1$. The stated flow condition $U_2L_2^2=2U_1L_1^2$ gives $$U_2=\dfrac{2U_1L_1^2}{L_2^2}=\dfrac{2U_1L_1^2}{4L_1^2}=\dfrac{U_1}{2}$$ Check $Re_d$: $Re_2=\rho U_2 d_2/\mu=\rho(U_1/2)(2d_1)/\mu=\rho U_1 d_1/\mu=Re_1$ — Reynolds similarity is automatically preserved by the given flow scaling. Since the gas/solid materials are unchanged, $Pr$ and $C_s/C_g$ also match, and $L/d$ matches by construction, so the only remaining condition for identical $\Delta\theta_{g,t}/\Delta\theta_0$ is matching $\Pi_5=U_0t/L$: $$\dfrac{U_1t_1}{L_1}=\dfrac{U_2t_2}{L_2}\ \Rightarrow\ \dfrac{t_2}{t_1}=\dfrac{U_1}{U_2}\cdot\dfrac{L_2}{L_1}=(2)(2)=\boxed{4}$$