For the barbequing and microwaving cases, some of these input parameters were chosen after the fact, to give cooling times consistent with experience. For barbequing, the fluid temperature is taken to be 500 °C. In practice, raising the distance between the coals and the food lowers the fluid temperature in contact with the food. There is an important radiation effect from the coals that is not considered here as it does not fit into either radiation model. For the microwaving case, the value of the rate of heat generation is given in Watts absorbed per unit volume, and the value (1(10)7 W/m3 ) is chosen to be one which gives a cooking time consistent with empirical experience. That’s cheating, a little… Or, rather, it is using experimental results and a theoretical model to evaluate an approximation for the effective rate of absorption of microwave energy, for which no attempt to model from first principles is made here.
Model Development A general 1-node model with a heat source (for the microwave case) and a thermal channel between the capacity-carrying node and ambient with a conductive and convective/radiative channel in series is shown in Fig.
4.39 .
Fig. 4.39 General 1-node model applicable to hot dog model
The energy balance applied to this general RC network is:
$$ C\frac{dT}{dt}=\dot{Q}+\frac{T_{\infty }-T}{R_k+{R}_h} $$
The solution (with constant parameter values) has been given in Chap.
1 :
$$ \frac{T-{T}_{SS}}{T_0-{T}_{SS}}={e}^{\frac{-t}{C\left({R}_k+{R}_h\right)}} $$
where
\( {T}_{ss}={T}_{\infty }+\dot{Q}\left({R}_k+{R}_h\right) \) is the steady-state temperature and T
0 is the initial temperature. Note that the microwave case is the only one for which there is internal heat generation, and there is a difference between the steady-state and ambient fluid temperatures.
The surface temperature is directly related to the average hot dog temperature through the voltage divider channel between the capacity-carrying node and the outer hot dog surface:
$$ \frac{T-{T}_{\infty }}{R_k+{R}_h}=\frac{T_s-{T}_{\infty }}{R_h}=\frac{T-{T}_s}{R_k} $$
$$ {T}_s={T}_{\infty }+\frac{R_h\left(T-{T}_{\infty}\right)}{R_k+{R}_h}=T-\frac{R_k\left(T-{T}_{\infty}\right)}{R_k+{R}_h} $$
While not essential, it is useful to define Biot and Fourier numbers:
$$ Bi={R}_k/{R}_h\kern0.5em \mathrm{Gives}\kern0.5em \mathrm{SPATIAL}\kern0.5em \mathrm{INFO}\kern1em Fo=t/C{R}_k\kern0.5em \mathrm{Gives}\kern0.5em \mathrm{TEMPORAL}\kern0.5em \mathrm{INFO} $$
$$ \frac{T-{T}_{SS}}{T_0-{T}_{SS}}={e}^{\frac{-t}{C{R}_k\left(1+\raisebox{1ex}{${R}_h$}\!\left/ \!\raisebox{-1ex}{${R}_k$}\right.\right)}}={e}^{-\left(\frac{Bi}{1+ Bi}\right)Fo} $$
$$ {T}_s={T}_{\infty }+\frac{\left(T-{T}_{\infty}\right)}{Bi+1}=T-\frac{Bi\left(T-{T}_{\infty}\right)}{Bi+1} $$
Consider the Biot number limits:
For Bi ≫ 1, there is good convection and Ts → T∞ , that is, the hot dog cooks on the edges first, and the center lags behind.
For Bi ≪ 1, there is good conduction, and Ts → Tave , that is, the hot dog cooks with a uniform temperature.
Evaluating the capacitance is unambiguous:
$$ C=\rho cL\pi {D}^2/4 $$
Similarly, evaluating the convective resistance is unambiguous because convection occurs at a single, specific surface:
$$ {R}_h=1/\left( h\pi DL\right) $$
The convection coefficient is considered to be an average value. For the barbequing case, it is known from experience that the hot dog cooks more on the bottom side. That effect is probably due to radiation heat coming from the hot coals, an effect that could be accounted for by a parallel radiation channel acting on the bottom surface (but from a different temperature than the hot gases flowing past it, requiring a new RC network be drawn). But think of the hot dog as being turned frequently during the heating, and the higher convection coefficient value takes the nonuniform heating effect into account.
Evaluation of the conductive resistance requires some more thought because conduction occurs between two surfaces that have different surface area. As suggested by the sketch in Fig.
4.40 , the cylindrical hot dog is modeled as an equivalent wall, with one “surface” (of zero surface area) being the centerline, and the other surface being the outer surface of the hot dog, with surface area
πDL (with L the length of the hot dog), a distance
D /2 away. The capacity-carrying node is placed at some radius between the centerline and the surface.
Fig. 4.40 Conceptual model for estimating hot dog conductive resistance
The general form of a conductive resistance is:
$$ {R}_k=\frac{1}{k}{\displaystyle \underset{x_1}{\overset{x_2}{\int }}\frac{dx}{A}} $$
which has the general form of an average distance between surfaces (Δx) divided by the thermal conductivity, and an average area. A few models (summarized in Table
4.4 ) are considered before obtaining results for the base case hot dog.
Table 4.4 Summary of conduction models for 1-node hot dog problem
MODEL 1: The most straightforward conduction model (for conductive resistance between the hot dog surface and the capacity-carrying node) is to “place” the capacity-carrying node at the centerline of the hot dog (at r = 0), and use the basic estimate for R k as the distance between surfaces dividing by the thermal conductivity times average surface area.
MODEL 2: The centerline in a cylindrical geometry acts like an insulated boundary because the surface area approaches zero with the radius. Conceptually, heat conducted from the surface into the interior does not require it to conduct all the way to the centerline. Therefore, placement of the capacity-carrying node at the centerline will overestimate the conductive resistance. This model places the capacity-carrying node at the midpoint between the surface and the centerline (at r = D/4). The conductive resistance is based on the distance between the surface and node, and on the average areas. When executed, the functional dependencies are the same as Model 1, but the Biot number is reduced by a factor of 2/3.
MODEL 3: This model places the capacity-carrying node at the centroid of a wedge of the hot dog (at r = D/3), rather than at the midpoint between the centerline and center. The centroid is more representative of the solid than the geometric center. The Biot number is 2/5 of the value of Model 1.
MODEL 4: The model places the capacity-carrying node at the centroid of a wedge, but uses the exact formula for conduction between two surfaces at different radial positions, that is, across a cylindrical shell. The Biot number is reduced by a factor of 4.93 compared to Model 1.
Model 4 is considered to be conceptually the most accurate for this case. However, the important take-home message is that all models give the same functional dependency. Moving forward, methods for breaking the hot dog into multiple nodes will be developed, and the differences between models for conduction will be less important.
MODEL 4 results:
Spreadsheet results using conduction Model 4 are summarized in Fig.
4.41 and plots of the average hot dog temperature and the surface temperature are shown in Figs.
4.42 ,
4.43 ,
4.44 , and
4.45 . The conductive resistance and capacitance are independent of the method of cooking and are calculated in the top section (derived parameters) of a worksheet. This block refers to the input parameter block of the problem statement. The convective resistance depends on the method of cooking. Cooking in boiling water results in a large (but not effectively infinite) Biot number (4.73) event, which means conduction through the hot dog is slow compared to the rate at which heat is transferred by convection from the water to the surface. There is a large spatial gradient, with the surface temperature much closer to the fluid temperature than the core temperature. The other three cases are low Biot number cases (but not effectively zero), which means that conduction is effective relative to convection and the hot dog cooks nearly uniformly. Comparing the boiling and baking cases, the cooking time is faster in boiling water, despite the fact that the fluid temperature (100 °C) is lower than that in an oven (177 °C). But at the time the hot dog is cooked, the surface temperature in water is 95 °C (25 °C above the hot dog average, and only 5 °C below the fluid temperature) while the surface temperature is 79 °C for baking (9 °C above the hot dog, and almost 100 °C below the fluid temperature).
Fig. 4.41 Results of 1-node hot dog workshop for conduction Model 4
Fig. 4.42 Boiling: Biot number = 4.73, Tinf = 100 °C
Fig. 4.43 Baking: Biot number = 0.089, Tinf = 177 °C
Fig. 4.44 Barbeque, Biot number = 0.148, Tinf = 500 °C
Fig. 4.45 Microwave, Biot number = 0.089, Tinf = 25 °C
For barbequing, the cooking time is shorter than boiling, because the fluid temperature (500 °C) is so high. Meanwhile, the surface temperature is 125 °C. In this case, despite the low Biot number, the hot dog appears to heat nonuniformly, but that is merely because the fluid temperature is so high. The surface temperature is closer to the average hot dog temperature than it is to the fluid temperature. The fluid temperature chosen for this case is not well characterized. Nevertheless, it is well known from experience that it is a function of the vertical distance between the source of the fire and the hot dog. It burns on the outside if placed too close.
Finally, microwaving yields the shortest cooking times (but to me, a hot dog in a microwave is too rubbery…). The equilibrium (steady-state) temperature of 3,200 °C represents the temperature the hot dog would achieve in theory if the microwave were left on indefinitely. Of course, something would happen to the hot dog to change the model long before it reached that point.
The surface temperature at the microwave cooking time is lower than surface temperature. That is, heat flows outward from the core of the hot dog to the surface. Heat is absorbed uniformly throughout the interior, yet the temperature is not uniform. In order for heat to flow from the interior to the surface, and ultimately to the ambient, there must be a temperature gradient driving it. From Fig. 4.45