Opening the basement door does not really help to heat the basement - the hotter the air gets upstairs, the larger the difference in density gets, compared to the air in the basement.
So, convection will not work (- even with turbulences, slight wind - the hot air needs to move some meters down, so that all would not help).
Conduction through the floor will get stronger/more effective with higher temperature difference - so that would help heating at least a little.
Theoretically, opening the door could allow some heating of the basement by radiation from upstairs, if the geometry is suitable.
That would not have a strong effect, of course.
In total, opening the door does not do much, and the heating of the basement is mainly by conduction.
An interesting aspect is that the increased temperature difference will result in a stronger separation of hot and cold air.
That may reduce heat flow by diffusion - not sure.
From a practical perspective, it may be relevant that the air in the basement is heated from the ceiling, so much of the heat will end up in a layer of warmer air there.
So first off, i'm going to have to break to you that this $\Delta E_\text{cond}(t) + \Delta E_\text{conv}(t)$
is by far the most complicated bit.
There is a simple equation $\frac{q}{A} = h\Delta T$
where $q$ is the heat flux (W/m^2), $A$ is the surface area of the object, $\Delta T$ is the temperature difference between the bulk of the fluid and the object and $h$ is the heat transfer coefficient.
Unfortunately $h$ is quite difficult to determine as it depends on the properties of the fluid and its flow regime as well as the geometry of the problem.
In order to determine the heat transfer coefficient we define a number of dimensionless quantities such as $Pr$; the Prandtl number. This is the ratio of momentum diffusivity to thermal diffusivity for the fluid. Effectively this describes the relative size of the thermal and velocity boundary layers for the fluid. For low $Pr$ conduction through the fluid dominates (Mercury has Pr of around 0.015). In the opposite case of engine oil momentum diffusivity is much larger so convection dominates (Pr of 100 to 40000). The Prandtl number for air and most other gases is around 0.7-0.8.
Obviously the Reynolds number for the fluid, $\Re _\text{L}$ is going to be important and is defined as $\Re = \frac{v L}{\nu}$
This describes the flow regime around the object where $L$ is a characteristic length and can be taken to be the diameter of the sphere in this case. $\nu$ is the kinematic viscosity and $v$ is the velocity of the object relative to the bulk of the fluid. For the appropriate substance and temperature you can find the kinematic viscosity of your gas and then calculate the appropriate Reynolds number.
Lastly we define the Nusselt number. $Nu _\text{L} = \frac{h L}{k}$
where $L$ is again the characteristic length, $h$ is the heat transfer coefficient and $k$ is the thermal conductivity of the fluid which is evaluated at the film temperature. This is defined as the arithmetic mean of the temperature of the bulk of the fluid $T _\text{g}$ also known as $T _\infty$ and the temperature at the solid boundary.
As a small aside I will mention the Biot number. $Bi = \frac{h L}{k _\text{b}}$
where the symbols have their usual meanings except $k _\text{b}$ is the thermal conductivity of the body (your sphere in this case). If this is 0.1 or less then the conduction throughout the solid body is sufficient to assume that the body has no internal temperature gradient (which simplifies your problem).
Now on to solving the problem. We want to express the Nusselt number in terms of the Prandtl and Reynolds numbers in order to find an expression for $h$.
Unfortunately this is very difficult but luckily large cohorts of experimentalists have slaved away at this problem for us.
For external flow over a sphere, T. Yuge (I don't have an explicit reference for this see textbook link at the end) found the following correlation $Nu = 2 + 0.43 Re ^\frac{1}{4} $.
This is for Pr approximately equal to 1 and Reynolds number between 1 and 10000.
(I could not find another correlation when Pr is closer to the range expected for air and most other gases. You will have to either accept this approximation or find your own information sorry.)
Decide on appropriate characteristics of your fluid and solid. Calculate the Reynolds and hence Nusselt numbers and then the heat transfer coefficient. Use this to solve for the heat flux for an appropriate temperature difference and then put this in your integral with your radiation term to solve for the total change in energy. I think that about does it. Here is a textbook that goes into a lot of detail:
http://www.unimasr.net/ums/upload/files/2012/Sep/UniMasr.com_919e27ecea47b46d74dd7e268097b653.pdf
I hope this is helpful.
Best Answer
The reason for this is actually the temperature-dependent distribution of the absolute value of particle velocities (higher temperature results in higher average velocities as well as greater variance of the underlying distribution) and the impact forces will have on particles with different velocities. The buoyancy is not a result of a pressure force (it does not emerge from a pressure term) but instead it emerges directly from gravity acting upon the gas molecules moving at lower or higher speed. On a macroscopic level this turns out to be correlated to the local density. I will try to illustrate this in the following section by adapting the macroscopic (Archimedes principle and Navier-Stokes-Fourier equations) and the microscopic view (Boltzmann equation) of physics.
My post turned out quite lengthy. I am pretty sure one could give a more concise answer but after having put a lot of time into writing it I do not feel like deleting most of it again.
You have probably already heard the saying (commonly attributed to the Brit George Box) "All models are wrong, but some are useful": Models fall short of the complexity of real physics but some can be used to make valuable predictions if somebody is aware of the limits and pitfalls. Maybe some things cannot be described by a model because it is too coarse to capture the phenomena (it does not resolve the scale of interest) and some models emerge as special cases from other models.
In the case of macroscopic fluid dynamics of gases (as a matter of fact most common fluids) the macroscopic description is given by a set of partial differential equations in these macroscopic variables, the Navier-Stokes-Fourier equations. The description assumes that a gas is so dense that one can find limit values for macroscopic properties like density, stresses or specific forces
$$\rho := \lim\limits_{\Delta V \rightarrow 0} \frac{\Delta m}{\Delta V}, \phantom{abcdefg} \sigma_{ij} := \lim\limits_{\Delta A_i \rightarrow 0} \frac{\Delta F_j}{\Delta A_i}, \phantom{abcdefg} g_i := \lim\limits_{\Delta m \rightarrow 0} \frac{\Delta G_i}{\Delta m}$$
as well as temperature $T$, an assumption commonly referred the continuum assumption. This breaks down for very dilute gases, where diluteness (generally specified by $Kn \gtrsim 0.01$) is a question of the scale of interest, generally characterised by the Knudsen number $Kn$ which defines that ratio between the mean free path $\lambda$ and the characteristic length scale of the problem $L$:
$$ Kn := \frac{\lambda}{L} \phantom{spacespace} \frac{\text{mean free path}}{\text{representative physical length scale}}$$
This means that something that might seem dense on a larger scale might be too dilute on a smaller scale to be considered a continuum after all.
The macroscopic description can be derived as the limit case for dense gases from a microscopic (or rather mesoscopic) description from kinetic theory of gases, the Boltzmann equation, in the limit of dense gases by a perturbation analysis, the Chapman-Enskog expansion. One might argue that the Navier-Stokes equations emerge as a special from the more general Boltzmann equation and that the Boltzmann equation is more general. They are two different levels of description and can be used to explain the same phenomena, while the reason for certain phenomena cannot be fully explained on the coarse macroscopic level. I would argue that advection, diffusion and heat conduction are all actually macroscopic phenomena: They are caused by microscopic mechanisms but are actually only visible on a much larger scale.
Now we are going to investigate the buoyancy on these two levels of description for your two different cases and see how they two are different.
1. Buoyancy on the macroscopic level
1.1 With the balloon
Already 200 B.C. Archimedes, a greek mathematician and physicist, found out that static buoyancy is connected to a difference in density (Archimedes' principle):
A difference in density and the resulting displacement will result in a force: A balloon filled with hot air having a lower density than the surrounding cold air will result in a net force upwards. This means neglecting friction (viscous effects) one could actually easily calculate the resulting force. This can be expressed in mathematical more rigorous terms by integration of stresses:
$$ F = - \int\limits_{0}^{\pi} \sigma_{r \theta} |_{r=R} \sin(\theta) dA + \int\limits_{0}^{\pi} \sigma_{r r} |_{r=R} \cos(\theta) dA $$
If you are able to determine the distribution of stresses $\sigma_{r \theta}$ and $\sigma_{r r}$ acting inside the balloon and outside of it, you can determine the resulting force.
For the static case this is pretty straight forward while for any dynamic case this can be pretty complicated as one would have to consider the full Navier-Stokes-Fourier equations (continuity equation \eqref{1}, momentum equation \eqref{2}, energy equation \eqref{3}, Fourier's law \eqref{4} and the constitutive equation \eqref{5} as well as the equation of state \eqref{6}) and can only be done for very simple symmetric cases in laminar flows.
$$\frac{\partial \rho}{\partial t} + \sum\limits_{j \in \mathcal{D}} \frac{\partial (\rho u_j )}{\partial x_j }=0 \tag{1}\label{1}$$
$$\frac{\partial (\rho u_i )}{\partial t} + \sum\limits_{j \in \mathcal{D}}\frac{\partial (\rho u_i u_j )}{\partial x_j} = \sum\limits_{j \in \mathcal{D}} \frac{\partial \sigma_{ij}}{\partial x_j } + \rho g_i \tag{2}\label{2}$$
$$\frac{\partial (\rho e)}{\partial t} + \sum\limits_{j \in \mathcal{D}} \frac{\partial (\rho u_j e)}{\partial x_j} = - \sum\limits_{j \in \mathcal{D}} \frac{\partial q_j}{\partial x_j} + \sum\limits_{(i, j) \in \mathcal{D}} \frac{\partial (\sigma _{ij} u_i)}{\partial x_j} + \sum\limits_{j \in \mathcal{D}} \rho u_j g_j \tag{3}\label{3}$$
$$q_i = - k \frac{\partial T}{\partial x_i}. \tag{4}\label{4}$$
$$\sigma_{ij} = - p \delta_{ij} + 2 \mu S_{ij} - \frac{2}{3} \mu \sum\limits_{k \in \mathcal{D}} S_{kk} \delta_{ij}. \tag{5}\label{5}$$
$$p \, v = \frac{p}{\rho} = R_m T \tag{6}\label{6}$$
where $S_{ij}$ is the rate of strain tensor
$$S_{ij} := \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right).$$
If you are assuming a sperical body that is only falling or rising slowly you can actually neglect most of the terms and simplify the equations to determine the terminal sinking speed analytically by extending the derivation here with a simple force balance. The most important equation in this context is the momentum equation \eqref{2} which is a continuum equivalent of Newton's second law.
In your case with the hot and cold air and the elastic membrane the two would push the membrane such that there is an equilibrium between the two with equal pressure. If we assume that the membran of the balloon is adiabatic (and further assuming density differences that are sufficiently small such that the terminal speed is still laminar in the creeping regime) then the formulas mentioned in the post above would hold.
1.2 Without the balloon
Without the balloon other terms of the Navier-Stokes-Fourier equation system will gain importance. The conservation equations in the end are nothing but advection-diffusion equations with different diffusivities of the following form:
$$\underbrace{\frac{\partial \Phi_i}{\partial t}}_{\text{temporal change}} + \underbrace{\sum\limits_{j \in \mathcal{D}} \frac{\partial (\Phi_i u_j )}{\partial x_j }}_{\text{change due to advection}} = \underbrace{ \sum\limits_{j \in \mathcal{D}} \frac{\partial D_i}{\partial x_j } }_{\text{diffusion}} + \underbrace{S_i}_{\text{source}}$$
As you can see from \eqref{2} and the equation above the gravity force term is not part of the pressure term - instead it is a source term.
Now if there is no balloon the air of different temperature can mix (clearly if they were two kinds of different fluids and the coherence between the fluid of the same kind would too large the two would not). This means that the differences between the hot and the cold air will even out over time due to the diffusive term. You would still have the buoyancy but it would interfere with diffusion of momentum and energy. If you wait sufficiently enough the air would have the same temperature and density. Which mechanism dominates depends on the difference in temperature and the corresponding diffusivities.
I want to make a brief remark at this point on the post that sparked your question: It is true that in natural convection the pressure of the two hot and the cold air is actually identical. Pressure disturbances propagate with the speed of sound
$$ c_s := \sqrt{\left( \frac{\partial p}{\partial \rho} \right)_S } = \sqrt{\gamma R_m T} $$
This means that for phenomena that involve fluid flow at lower speeds (characterised by a Mach number $Ma := U / c_s$ lower than 1) the flow will be aware of the increase in pressure, resulting in a smooth change in pressure between two reservoires of hot and cold air. For supersonic flow ($Ma > 1$, e.g. a balloon with a sufficiently high pressure difference to the surrounding medium and a de Laval nozzle) this is different and there would be a difference in pressure between the two streams.
2. Kinetic theory of gases
Now if we want to know what happens on a level of gas molecules we will have to turn to the kinetic theory of gases. For this we assume a simple model gas made of billiard-ball-like molecules that interact with each other in elastic collisions. These collisions preserve mass, momentum and energy. Contrary to your assumptions and as already outlined in my previous answer, collisions in gases are not rare but actually pretty common due to the immense number of particles. While in a real gas you will have far-field interactions due to repulsive forces, it allows us to get quite a good approximation.
2.1 Boltzmann equation
Properties like the density $\rho ( \vec x, t )$ are a measure for the average state of particles. When considering dilute gases a density cannot be found but instead one may find particles at a certain point in space $\vec x$ at a certain time $t$ with a corresponding velocity $\vec \xi$ with a corresponding probability. The basis of kinetic theory is the introduction of a particle distribution function (which can be seen as a generalisation of the density $\rho$, which is only a function of time and space $\rho(\vec x,t)$)
$$f(\vec x, \vec \xi, t) := \frac{dN}{d \vec x \, d \vec \xi}$$
where $N$ denotes the number of molecules.
Now all relevant macroscopic variables can be determined as moments of the distribution function through integration over the velocity space. The zeroth moment equals to the density
$$ \rho(\vec x, t) = m_P \int f(\vec x, \vec \xi, t) d \vec \xi, $$ the first moment to the impuse
$$\rho(\vec x, t) \vec u(\vec x, t) = m_P \int \vec \xi f(\vec x, \vec \xi, t) d \vec \xi, $$
where $\vec u$ is the mean macroscopic velocity. The second moment delivers the total energy density
$$ \rho(\vec x, t) e(\vec x, t) = \frac{m_P}{2} \int \vec \xi^2 f(\vec x, \vec \xi, t) d \vec \xi, $$
Now we want to determine how this distribution evolves in time, the so called Boltzmann transport equation. In a real system an ensemble contains theoretically information about its state at any time but due to the enormous number of interactions over time this is converted into subtle correlations that appear chaotic (The time over which the system may be predicted is very limited.) and almost random. When deriving this equation Boltzmann therefore assumed that the particles are un-correlated before collisions (one-sided molecular chaos). In dilute gasses it can be further assumed that the particles ($i=1, \ldots , N$) spend most of their lifespan on free trajectories apart from collisions involving only two particles (Collisions involving multiple particles are so rare they can be neglected.) at a time for which in the case of mono-atomic (billiard balls) gas classical Newtonian physics can be assumed
$$\frac{d x_i}{dt}= \xi_i = \frac{P_i}{m_P}, \phantom{space} \frac{d P_i}{dt} = F_i.$$
where $\vec P$ stands for the transferred momentum. In such a model the particle distribution would follow a simple transport equation, the Boltzmann transport equation. The total derivative of a distribution function $f$ given by
$$\left. \frac{Df}{Dt} \right\vert_{transport} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial x_i} \underbrace{ \frac{\partial x_i}{\partial t} }_{\xi_i} + \frac{\partial f}{\partial \xi_i} \underbrace{ \frac{\partial \xi_i}{\partial t} }_{\frac{F_i}{m_P}}. $$
must be equal to the changes caused by the collision. This can be rewritten using $\partial x_i/\partial t=\xi_i$ and $\partial \xi_i/\partial t=F_i/m$ as well as introducing the notation $\Omega(f)=Df/Dt$ for the not yet defined collision integral to
$$\underbrace{ \frac{\partial f}{\partial t} + \vec \xi \boldsymbol{\cdot} \vec \nabla f + \frac{\vec F}{m_P} \boldsymbol{\cdot} \vec \nabla_{\vec \xi} f }_\text{Propagation} = \underbrace{ \Omega(f) }_\text{Collision}$$
Under certain assumptions one can derive such a collision operator and simplified versions of it, such as the BGK, and interestingly by considering symmetries we can find a peculiar distribution that describes $f$ in the equilibrium state $f$ as sketched out over here.
Again you can see there is an explicit body force term, as we will see this is again the source of the body force on a macroscopic level and therefore of our buoyancy.
Now we could ask ourselves what happens on the level of density, momentum and energy on a much larger scale. This can be done by calculating the aforementioned moments of the Boltzmann equation. Now certainly we do not know what the distribution $f$ looks like. The easiest way of doing so is by assuming an equilibrium distribution. In equilibrium regardless of the precise form of the collision integral it has to be orthogonal to any collision invariant, meaning it has to preserve mass, momentum and energy. When doing so, you end up with the Euler equations for inviscid flow.
When assuming a perturbation from the equilibrium, with a perturbation series of the form
$$f = \sum_{n=0}^{\infty} \epsilon^n f^{(n)}$$
one has to evaluate the integrals of the collision operator and the whole things becomes quite complicated, a procedure known as the Chapman-Enskog expansion, and one ends up with the full Navier Stokes equations in the limit of small Knudsen numbers.
Anyways the interesting term here for us is the force term in the momentum equation. Let us take the first moment (which results in the momentum equation) of the force term
$$ m_P \int \xi_i \, \underbrace{\frac{\vec F}{m_P}}_{\vec g} \boldsymbol{\cdot} \vec \nabla_{\vec \xi} f \, d \vec \xi $$
alone. With integration by parts and using $\int \frac{\partial f}{\partial \xi_i} d \vec \xi= 0$ we get
$$ m_P \int \xi_j \frac{\partial f}{\partial \xi_i} d \vec \xi= - m_P \int \frac{\partial \xi_j}{\partial \xi_i} f d \vec \xi = - \rho \delta_{i j} $$
and as a result first moment of the forcing term on the left-hand side of the Boltzmann equation $m_P \int \xi_i \, \vec g \boldsymbol{\cdot} \vec \nabla_{\vec \xi} f \, d \vec \xi$ becomes the gravity term $\rho g_i$ on the right-hand-side of the macroscopic momentum equation.
The force term in the Navier Stokes equation completely emerges from the force term in the Boltzmann equation, it does not contribute to pressure there either. Basically when moving faster on a macroscopic level (as the faster molecules of the hotter gas do) the effects of gravity are less apparent and somehow the resulting term is proportional to the density on a macroscopic level.
So now we have all the tools to explain the effects on a microscopic level, which is quite similar to the macroscopic case.
2.2 With the balloon
Our molecules to both sides of the membrane, the ones belonging to the hot and the cold air, will collide with it. They will displace the membrane until there is an equilibrium in pressure to both sides. The density on the hot side will be smaller than on the cold side as the equilibrium means that on that side atoms are more apart and less densely packed. This can also be seen from the mean free path which can be calculated in kinetic theory (again with the same assumptions as above) to
$$\lambda = \frac{m_P}{\sqrt{2} \pi d^2 \rho}.$$
2.3 Without the balloon
Without the balloon we will start with two distributions that are different between hot and cold air. They will not be an equilibrium distribution in each case but something in between the two distributions. Every collision between molecules of the two distributions will even out this difference and drive the gas locally to equilibrium. This equilibrated molecules will collide with their neighbours resulting in an diffusive layer that spreads out and blurs the interface between the two. The diffusivity for the momentum is the viscosity which we can estimate to
$$ \mu = \frac{ 5\sqrt{\pi}}{16} \frac{\sqrt{k_B m_P T}}{\pi d^2} $$
As pointed out before the gravity will have less of an effect of the molecules travelling on higher speeds, while so to speak a more prominent effect on slower moving molecules, which happens to be proportional to the local density. These two effects are acting on the same time resulting in a blurring of the interface between the hot and cold air due to diffusion and to a rising motion of the hot air due to the density differences.